?Normal lines A normal line to a curve passes through a point P on the curve

For some equations, such as \(x^{2}+y^{2}=1\) or \(x-y^{2}=0\), it is possible to solve for y and then calculate \(\frac{d y}{d x}\). Even in these cases, explain why implicit differentiation is usually a more efficient method for calculating the derivative.

Explain the differences between computing the derivatives of functions that are defined implicitly and explicitly.

Why are both the x-coordinate and the y-coordinate generally needed to find the slope of the tangent line at a point for an implicitly defined function?

In this section, for what values of n did we prove that \(\frac{d}{d x}\left(x^{n}\right)=n x^{n-1}\)?

Implicit differentiation Carry out the following steps. a. Use implicit differentiation to find the derivative \(\frac{d y}{d x}\). b. Find the slope of the curve at the given point. \(y^{2}=4 x ;(1,2)\)

Implicit differentiation Carry out the following steps. a. Use implicit differentiation to find the derivative \(\frac{d y}{d x}\). b. Find the slope of the curve at the given point. \(y^{2}+3 x=2 ;(-1, \sqrt{5})\)

Implicit differentiation Carry out the following steps. a. Use implicit differentiation to find the derivative \(\frac{d y}{d x}\). b. Find the slope of the curve at the given point. \(\sin (y)=5 x^{4}-5 ;(1, \pi)\)

Implicit differentiation Carry out the following steps. a. Use implicit differentiation to find the derivative \(\frac{d y}{d x}\). b. Find the slope of the curve at the given point. \(5 \sqrt{x}-10 \sqrt{y}=\sin x ;(4 \pi, \pi)\)

Implicit differentiation Carry out the following steps. a. Use implicit differentiation to find the derivative \(\frac{d y}{d x}\). b. Find the slope of the curve at the given point. \(\cos (y)=x ;\left(0, \frac{\pi}{2}\right)\)

Implicit differentiation Carry out the following steps. a. Use implicit differentiation to find the derivative \(\frac{d y}{d x}\). b. Find the slope of the curve at the given point. tan (xy) = x + y; (0, 0)

Implicit differentiation Use implicit differentiation to find \(\frac{d y}{d x}\). sin (xy) = x + y

Implicit differentiation Use implicit differentiation to find \(\frac{d y}{d x}\). \(e^{x y}=2 y\)

Implicit differentiation Use implicit differentiation to find \(\frac{d y}{d x}\). \(\cos \left(y^{2}\right)+x=e^{y}\)

Implicit differentiation Use implicit differentiation to find \(\frac{d y}{d x}\). \(y=\frac{x+1}{y-1}\)

Implicit differentiation Use implicit differentiation to find \(\frac{d y}{d x}\). \(x^{3}=\frac{x+y}{x-y}\)

Implicit differentiation Use implicit differentiation to find \(\frac{d y}{d x}\). \((x y+1)^{3}=x-y^{2}+8\)

Implicit differentiation Use implicit differentiation to find \(\frac{d y}{d x}\). \(6 x^{3}+7 y^{3}=13 x y\)

Implicit differentiation Use implicit differentiation to find \(\frac{d y}{d x}\). \(\left(x^{2}+y^{2}\right)\left(x^{2}+y^{2}+x\right)=8 x y^{2}\)

Implicit differentiation Use implicit differentiation to find \(\frac{d y}{d x}\). \(\sqrt{x^{4}+y^{2}}=5 x+2 y^{3}\)

Implicit differentiation Use implicit differentiation to find \(\frac{d y}{d x}\). \(\sqrt{3 x^{7}+y^{2}}=\sin ^{2} y+100 x y\)

Tangent lines Carry out the following steps. a. Verify that the given point lies on the curve. b. Determine an equation of the line tangent to the curve at the given point. \(x^{2}+x y+y^{2}=7 ;(2,1)\)

Tangent lines Carry out the following steps. a. Verify that the given point lies on the curve. b. Determine an equation of the line tangent to the curve at the given point. \(x^{4}-x^{2} y+y^{4}=1 ;(-1,1)\)

Tangent lines Carry out the following steps. a. Verify that the given point lies on the curve. b. Determine an equation of the line tangent to the curve at the given point. \(\sin y+5 x=y^{2} ;\left(\frac{\pi^{2}}{5}, \pi\right)\)

Tangent lines Carry out the following steps. a. Verify that the given point lies on the curve. b. Determine an equation of the line tangent to the curve at the given point. \(x^{3}+y^{3}=2 x y ;(1,1)\)

Tangent lines Carry out the following steps. a. Verify that the given point lies on the curve. b. Determine an equation of the line tangent to the curve at the given point. \(\cos (x-y)+\sin y=\sqrt{2}; \left(\frac{\pi}{2} \frac{\pi}{4}\right)\)

Tangent lines Carry out the following steps. a. Verify that the given point lies on the curve. b. Determine an equation of the line tangent to the curve at the given point. \(\left(x^{2}+y^{2}\right)^{2}=\frac{25}{4} x y^{2}; (1,2)\)

Second derivatives Find \(\frac{d^{2} y}{d x^{2}}\). \(x+y^{2}=1\)

Second derivatives Find \(\frac{d^{2} y}{d x^{2}}\). \(2 x^{2}+y^{2}=4\)

Second derivatives Find \(\frac{d^{2} y}{d x^{2}}\). \(\sqrt{y}+x y=1\)

Second derivatives Find \(\frac{d^{2} y}{d x^{2}}\). \(x^{4}+y^{4}=64\)

Second derivatives Find \(\frac{d^{2} y}{d x^{2}}\). \(e^{2 y}+x=y\)

Second derivatives Find \(\frac{d^{2} y}{d x^{2}}\). \(\sin x+x^{2} y=10\)

Derivatives of functions with rational exponents Find \(\frac{d y}{d x}\). \(y=x^{5 / 4}\)

Derivatives of functions with rational exponents Find \(\frac{d y}{d x}\). \(y=\sqrt[3]{x^{2}-x+1}\)

Derivatives of functions with rational exponents Find \(\frac{d y}{d x}\). \(y=(5 x+1)^{2 / 3}\)

Derivatives of functions with rational exponents Find \(\frac{d y}{d x}\). \(y=e^{x} \sqrt{x^{3}}\)

Derivatives of functions with rational exponents Find \(\frac{d y}{d x}\). \(y=\sqrt[4]{\frac{2 x}{4 x-3}}\)

Derivatives of functions with rational exponents Find \(\frac{d y}{d x}\). \(y=(2 x+3)^{2}(4 x+6)^{1 / 4}\)

Derivatives of functions with rational exponents Find \(\frac{d y}{d x}\). \(y=x \sqrt[3]{x^{2}+5 x+1}\)

Derivatives of functions with rational exponents Find \(\frac{d y}{d x}\). \(y=\frac{x}{\sqrt[5]{x}+x}\)

Implicit differentiation with rational exponents Determine the slope of the following curves at the given point. \(\sqrt[3]{x}+\sqrt[3]{y^{4}}=2 ;(1,1)\)

Implicit differentiation with rational exponents Determine the slope of the following curves at the given point. \(x^{2 / 3}+y^{2 / 3}=2 ;(1,1)\)

Implicit differentiation with rational exponents Determine the slope of the following curves at the given point. \((x+y)^{2 / 3}=y ;(4,4)\)

Implicit differentiation with rational exponents Determine the slope of the following curves at the given point. \(x y+x^{3 / 2} y^{-1 / 2}=2 ;(1,1)\)

Implicit differentiation with rational exponents Determine the slope of the following curves at the given point. \(x y^{5 / 2}+x^{3 / 2} y=12 ;(4,1)\)

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. For any equation containing the variables x and y, the derivative dy/dx can be found by first using algebra to rewrite the equation in the form y = f(x). b. For the equation of a circle of radius r, \(x^{2}+y^{2}=r^{2}\), we have \(\frac{d y}{d x}=-\frac{x}{y}\) for \(y \neq 0\) and any real number r > 0. c. If x = 1, then by implicit differentiation, 1 = 0. d. If xy = 1, then \(y^{\prime}=1 / x\).

Multiple tangent lines Complete the following steps. a. Find equations of all lines tangent to the curve at the given value of x. b. Graph the tangent lines on the given graph. \(x+y^{3}-y=1 ; x=1\)

Multiple tangent lines Complete the following steps. a. Find equations of all lines tangent to the curve at the given value of x. b. Graph the tangent lines on the given graph. \(x+y^{2}-y=1 ; x=1\)

Multiple tangent lines Complete the following steps. a. Find equations of all lines tangent to the curve at the given value of x. b. Graph the tangent lines on the given graph. \(4 x^{3}=y^{2}(4-x) ; x=2\) (cissoid of Diocles)

Multiple tangent lines Let \(y\left(x^{2}+4\right)=8\) (witch of Agnesi). a. Use implicit differentiation to find \(\frac{d y}{d x}\). b. Find equations of all lines tangent to the curve \(y\left(x^{2}+4\right)=8\) when y = 1. c. Solve the equation \(y\left(x^{2}+4\right)=8\) for y to find an explicit expression for y and then calculate \(\frac{d y}{d x}\). d. Verify that the results of parts (a) and (c) are consistent.

Vertical tangent lines a. Determine the points where the curve \(x+y^{3}-y=1\) has a vertical tangent line (see Exercise 48). b. Does the curve have any horizontal tangent lines? Explain.

Vertical tangent lines a. Determine the points where the curve \(x+y^{2}-y=1\) has a vertical tangent line (see Exercise 49). b. Does the curve have any horizontal tangent lines? Explain.

Identifying functions from an equation The following equations implicitly define one or more functions. a. Find \(\frac{d y}{d x}\) using implicit differentiation. b. Solve the given equation for y to identify the implicitly defined functions \(y=f_{1}(x), y=f_{2}(x), \ldots\) c. Use the functions found in part (b) to graph the given equation. d. Find the derivative of each function in part (b) and verify that your results are consistent with part (a). \(y^{3}=a x^{2}\) (Neile's semicubical parabola)

Identifying functions from an equation The following equations implicitly define one or more functions. a. Find \(\frac{d y}{d x}\) using implicit differentiation. b. Solve the given equation for y to identify the implicitly defined functions \(y=f_{1}(x), y=f_{2}(x), \ldots\) c. Use the functions found in part (b) to graph the given equation. d. Find the derivative of each function in part (b) and verify that your results are consistent with part (a). \(x+y^{3}-x y=1\) (Hint: Rewrite as \(y^{3}-1=x y-x\) and then factor both sides.)

Identifying functions from an equation The following equations implicitly define one or more functions. a. Find \(\frac{d y}{d x}\) using implicit differentiation. b. Solve the given equation for y to identify the implicitly defined functions \(y=f_{1}(x), y=f_{2}(x), \ldots\) c. Use the functions found in part (b) to graph the given equation. d. Find the derivative of each function in part (b) and verify that your results are consistent with part (a). \(y^{2}=\frac{x^{2}(4-x)}{4+x}\) (right strophoid)

Identifying functions from an equation The following equations implicitly define one or more functions. a. Find \(\frac{d y}{d x}\) using implicit differentiation. b. Solve the given equation for y to identify the implicitly defined functions \(y=f_{1}(x), y=f_{2}(x), \ldots\) c. Use the functions found in part (b) to graph the given equation. d. Find the derivative of each function in part (b) and verify that your results are consistent with part (a). \(x^{4}=2\left(x^{2}-y^{2}\right)\) (eight curve)

Identifying functions from an equation The following equations implicitly define one or more functions. a. Find \(\frac{d y}{d x}\) using implicit differentiation. b. Solve the given equation for y to identify the implicitly defined functions \(y=f_{1}(x), y=f_{2}(x), \ldots\) c. Use the functions found in part (b) to graph the given equation. d. Find the derivative of each function in part (b) and verify that your results are consistent with part (a). \(y^{2}(x+2)=x^{2}(6-x)\) (trisectrix)

Normal lines A normal line to a curve passes through a point P on the curve perpendicular to the line tangent to the curve at P (see figure). Use the following equations and graphs to determine an equation of the normal line at the given point and illustrate your work by graphing the curve with the normal line. Exercise 21

Normal lines A normal line to a curve passes through a point P on the curve perpendicular to the line tangent to the curve at P (see figure). Use the following equations and graphs to determine an equation of the normal line at the given point and illustrate your work by graphing the curve with the normal line. Exercise 22

Normal lines A normal line to a curve passes through a point P on the curve perpendicular to the line tangent to the curve at P (see figure). Use the following equations and graphs to determine an equation of the normal line at the given point and illustrate your work by graphing the curve with the normal line. Exercise 23

Normal lines A normal line to a curve passes through a point P on the curve perpendicular to the line tangent to the curve at P (see figure). Use the following equations and graphs to determine an equation of the normal line at the given point and illustrate your work by graphing the curve with the normal line. Exercise 24

Normal lines A normal line to a curve passes through a point P on the curve perpendicular to the line tangent to the curve at P (see figure). Use the following equations and graphs to determine an equation of the normal line at the given point and illustrate your work by graphing the curve with the normal line. Exercise 25

Normal lines A normal line to a curve passes through a point P on the curve perpendicular to the line tangent to the curve at P (see figure). Use the following equations and graphs to determine an equation of the normal line at the given point and illustrate your work by graphing the curve with the normal line. Exercise 26

Visualizing tangent and normal lines a. Determine an equation of the tangent line and normal line at the given point \(\left(x_{0}, y_{0}\right)\) on the following curves. (See instructions for Exercises 59-64.) b. Graph the tangent and normal lines on the given graph. \(3 x^{3}+7 y^{3}=10 y;\) \(\left(x_{0}, y_{0}\right)=(1,1)\)

Visualizing tangent and normal lines a. Determine an equation of the tangent line and normal line at the given point \(\left(x_{0}, y_{0}\right)\) on the following curves. (See instructions for Exercises 59-64.) b. Graph the tangent and normal lines on the given graph. \(x^{4}=2 x^{2}+2 y^{2};\) \(\left(x_{0}, y_{0}\right)=(2,2)\) (kampyle of Eudoxus)

Visualizing tangent and normal lines a. Determine an equation of the tangent line and normal line at the given point \(\left(x_{0}, y_{0}\right)\) on the following curves. (See instructions for Exercises 59-64.) b. Graph the tangent and normal lines on the given graph. \(\left(x^{2}+y^{2}-2 x\right)^{2}=2\left(x^{2}+y^{2}\right);\) \(\left(x_{0}, y_{0}\right)=(2,2)\) (limaçon of Pascal)

Visualizing tangent and normal lines a. Determine an equation of the tangent line and normal line at the given point \(\left(x_{0}, y_{0}\right)\) on the following curves. (See instructions for Exercises 59-64.) b. Graph the tangent and normal lines on the given graph. \(\left(x^{2}+y^{2}\right)^{2}=\frac{25}{3}\left(x^{2}-y^{2}\right);\) \(\left(x_{0}, y_{0}\right)=(2,-1)\) (lemniscate of Bernoulli)

Cobb-Douglas production function The output of an economic system Q, subject to two inputs, such as labor L and capital K, is often modeled by the Cobb-Douglas production function \(Q=c L^{a} K^{b}\). When a + b = 1 the case is called constant returns to scale. Suppose Q = 1280, \( a=\frac{1}{3}\), \(b=\frac{2}{3}\), and c = 40. a. Find the rate of change of capital with respect to labor, dK/dL. b. Evaluate the derivative in part (a) with L = 8 and K = 64.

Surface area of a cone The lateral surface area of a cone of radius r and height h (the surface area excluding the base) is \(A=\pi r \sqrt{r^2+h^2}\). Find dr/dh for a cone with a lateral surface area of \(A=1500 \pi~\mathrm{cm^2}\). Evaluate this derivative when r = 30 cm and h = 40 cm.

Volume of a spherical cap Imagine slicing through a sphere with a plane (sheet of paper). The smaller piece produced is called a spherical cap. Its volume is \(V=\pi h^{2}(3 r-h) / 3\), where r is the radius of the sphere and h is the thickness of the cap. Find dr/dh for a sphere with a volume of \(5 \pi / 3 \mathrm{~m}^{3}\). Evaluate this derivative when r = 2 m and h = 1 m.

Volume of a torus The volume of a torus (doughnut or bagel) with an inner radius of a and an outer radius of b is \(V=\pi^{2}(b+a)(b-a)^{2} / 4\). Find db/da for a torus with a volume of \(64 \pi^{2} \mathrm{in}^{3}\). Evaluate this derivative when a = 6 in and b = 10 in.

Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals). A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas \(y=c x^{2}\) form orthogonal trajectories with the family of ellipses \(x^{2}+2 y^{2}=k\), where c and k are constants (see figure). Use implicit differentiation if needed to find dy/dx for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. \(y=m x ; x^{2}+y^{2}=a^{2}\), where m and a are constants

Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals). A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas \(y=c x^{2}\) form orthogonal trajectories with the family of ellipses \(x^{2}+2 y^{2}=k\), where c and k are constants (see figure). Use implicit differentiation if needed to find dy/dx for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. \(y=c x^{2} ; x^{2}+2 y^{2}=k\), where c and k are constants

Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals). A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas \(y=c x^{2}\) form orthogonal trajectories with the family of ellipses \(x^{2}+2 y^{2}=k\), where c and k are constants (see figure). Use implicit differentiation if needed to find dy/dx for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. \(x y=a ; x^{2}-y^{2}=b\), where a and b are constants

Implicit differentiation with rational exponents Determine the slope of the following curves at the given point. \(x y^{1 / 3}+y=10 ;(1,8)\)