?Product Rule for three functions Assume that f, g, and h are differentiable at x.a. Use

How do you find the derivative of the product of two functions that are differentiable at a point?

How do you find the derivative of the quotient of two functions that are differentiable at a point?

State the Extended Power Rule for differentiating \(x^n\). For what values of n does the rule apply?

Give two ways to differentiate \(f(x)=1/x^{10}\).

What is the derivative of \(y=e^{kx}\)? For what values of k does this rule apply?

Give two ways to differentiate \(f(x)=(x-3)(x^2+4)\).

Find the derivative of the following functions. \(f(x)=3x^4(2x^2-1)\)

Find the derivative of the following functions. \(g(x)=6x-2xe^x\)

Find the derivative of the following functions. \(h(x)=(5x^7+5x)(6x^3+3x^2+3)\)

Find the derivative of the following functions. \(f(x)=(1+\frac{1}{x^2})(x^2+1)\)

Find the derivative of the following functions. \(g(w)=e^w(w^3-1)\)

Find the derivative of the following functions. \(s(t)=4e^t\sqrt{t}\)

Derivatives by two different methods a. Use the Product Rule to find the derivative of the given function. Simplify your result. b. Find the derivative by expanding the product first. Verify that your answer agrees with part(a). f(x) = (x - 1)(3x + 4)

Derivatives by two different methods a. Use the Product Rule to find the derivative of the given function. Simplify your result. b. Find the derivative by expanding the product first. Verify that your answer agrees with part(a). \(y=(t^2+7t)(3t-4)\)

Derivatives by two different methods a. Use the Product Rule to find the derivative of the given function. Simplify your result. b. Find the derivative by expanding the product first. Verify that your answer agrees with part(a). \(g(y)=(3y^4-y^2)(y^2-4)\)

Derivatives by two different methods a. Use the Product Rule to find the derivative of the given function. Simplify your result. b. Find the derivative by expanding the product first. Verify that your answer agrees with part(a). \(h(z)=(z^3+4z^2+z)(z-1)\)

Find the derivative of the following functions. \(f(x)=\frac{x}{x+1}\)

Find the derivative of the following functions. \(f(x)=\frac{x^3-4x^2+x}{x-2}\)

Find the derivative of the following functions. \(y=(3t-1)(2t-2)^{-1}\)

Find the derivative of the following functions. \(h(w)=\frac{w^2-1}{w^2+1}\)

Find the derivative of the following functions. \(g(x)=\frac{e^x}{x^2-1}\)

Find the derivative of the following functions. \(y=(2\sqrt{x}-1)(4x+1)^{-1}\)

Derivatives by two different methods a. Use the Quotient Rule to find the derivative of the given function. Simplify your result. b. Find the derivative by first simplifying the function. Verify that your answer agrees with part(a). \(f(w)=\frac{w^3-w}{w}\)

Derivatives by two different methods a. Use the Quotient Rule to find the derivative of the given function. Simplify your result. b. Find the derivative by first simplifying the function. Verify that your answer agrees with part(a). \(y=\frac{12s^3-8s^2+12s}{4s}\)

Derivatives by two different methods a. Use the Quotient Rule to find the derivative of the given function. Simplify your result. b. Find the derivative by first simplifying the function. Verify that your answer agrees with part(a). \(y=\frac{x-a}{\sqrt{x}-\sqrt{a}}\); a is a positive constant.

Derivatives by two different methods a. Use the Quotient Rule to find the derivative of the given function. Simplify your result. b. Find the derivative by first simplifying the function. Verify that your answer agrees with part(a). \(y=\frac{x^2-2ax+a^2}{x-a}\); a is a constant.

Equations of tangent lines a. Find an equation of the line tangent to the given curve at x = a. b. Use a graphing utility to graph the curve and the tangent line on the same set of axes. \(y=\frac{x+5}{x-1};\ \ a=3\)

Equations of tangent lines a. Find an equation of the line tangent to the given curve at x = a. b. Use a graphing utility to graph the curve and the tangent line on the same set of axes. \(y=\frac{2x^2}{3x-1};\ \ a=1\)

Equations of tangent lines a. Find an equation of the line tangent to the given curve at x = a. b. Use a graphing utility to graph the curve and the tangent line on the same set of axes. \(y=xe^x;\ \ a=-1\)

Equations of tangent lines a. Find an equation of the line tangent to the given curve at x = a. b. Use a graphing utility to graph the curve and the tangent line on the same set of axes. \(y=\frac{e^x}{x};\ \ a=1\)

Find the derivative of the following functions. \(f(x)=3x^{-9}\)

Find the derivative of the following functions. \(y=\frac{4}{p^3}\)

Find the derivative of the following functions. \(g(t)=3t^2+\frac{6}{t^7}\)

Find the derivative of the following functions. \(y=\frac{w^4+5w^2+w}{w^2}\)

Find the derivative of the following functions. \(g(t)=\frac{t^3+3t^2+t}{t^3}\)

Find the derivative of the following functions. \(p(x)=\frac{4x^3+3x+1}{2x^5}\)

Compute the derivative of the following functions. \(f(x)=15e^{3x}\)

Compute the derivative of the following functions. \(y=3x^2-2x+e^{-2x}\)

Compute the derivative of the following functions. \(g(x)=\frac{x}{e^x}\)

Compute the derivative of the following functions. \(f(x)=(1-2x)e^{-x}\)

Compute the derivative of the following functions. \(y=\frac{2e^x+3e^{-x}}{3}\)

Compute the derivative of the following functions. \(A=\ 2500e^{0.075t}\)

Consider the following population functions. a. Find the instantaneous growth rate of the population for \(t\ \geq\ 0\). b. What is the instantaneous growth rate at t = 5 hr? c. At what time is the instantaneous growth rate the greatest? d. Evaluate and interpret \(\lim _{t \rightarrow \infty}\ p'(t)\). e. Use a graphing utility to graph the population and its growth rate for \(0\ \leq\ t\ \leq\ 200\). \(p(t)=\frac{200t}{t+2}\)

Consider the following population functions. a. Find the instantaneous growth rate of the population for \(t\ \geq\ 0\). b. What is the instantaneous growth rate at t = 5 hr? c. At what time is the instantaneous growth rate the greatest? d. Evaluate and interpret \(\lim _{t \rightarrow \infty}\ p'(t)\). e. Use a graphing utility to graph the population and its growth rate for \(0\ \leq\ t\ \leq\ 200\). \(p(t)=\frac{800}{1+7e^{-0.2t}}\)

The half-life of an antibiotic in the bloodstream is 10 hr. If an initial dose of 20 mg is administered, the quantity left after t hours is modeled by \(Q(t)=20e^{-0.0693t}\) for \(t\ \geq\ 0\). a. Find the instantaneous rate of change of the amount of antibiotic in the bloodstream for \(0\ \leq\ t\ \leq\ 10\ hr\). b. How fast is the amount of antibiotic changing at t = 0 hr? At t = 2 hr? c. Evaluate and interpret \(\lim _{t \rightarrow \infty}\ Q(t)\) and \(\lim _{t \rightarrow \infty}\ Q'(t)\).

A $200 investment in a savings account grows according to \(A(t)=200e^{0.0398t}\) for \(t\ \geq\ 0\). a. Find the balance of the account after 10 years. b. How fast is the account growing (in dollars/year) at 1 = 10 years? c. Use your answers to parts (a) and (b) to write the equation of the line tangent to the curve \(A=200e^{0.0398t}\) at the point (10, A(10)).

Let \(f(x)=xe^{2x}\). a. Find the values of x for which the slope of the curve y = f(x) is 0. b. Explain the meaning of your answer to part (a) in terms of the graph of f.

Let \(f(t)=100e^{-0.05t}\). a. Find the values of t for which the slope of the curve y = f(t) is -5. b. Does the graph of f have a horizontal tangent line?

Compute the derivative of the following functions. \(g(x)=\frac{x(3-x)}{2x^2}\)

Compute the derivative of the following functions. \(h(x)=\frac{(x-1)(2x^2-1)}{(x^3-1)}\)

Compute the derivative of the following functions. \(g(x)=\frac{4e^{x/4}}{(x^2+x)(1-x)}\)

Compute the derivative of the following functions. \(h(x)=\frac{(x+1)}{x^2e^x}\)

Determine whether the following statements are true and give an explanation or counterexample. a. The derivative \(\frac{d}{dx}\ (e^5)\) equals \(5\cdot e^4\). b. The Quotient Rule must be used to evaluate \(\frac{d}{dx}\ (\frac{x^2+3x+2}{x})\) c. \(\frac{d}{dx}\ (\frac{1}{x^5})=\frac{1}{5x^4}\) d. \(\frac{d^n}{dx^n}(e^{3x})=3^n\cdot e^{3x}\) for any integer \(n\ \geq\ 1\).

Find f’(x), f’’(x), and f’’’(x). \(f(x)=\frac{1}{x}\)

Find f’(x), f’’(x), and f’’’(x). \(f(x)=x^2e^{3x}\)

Find f’(x), f’’(x), and f’’’(x). \(f(x)=\frac{x}{x+2}\)

Find f’(x), f’’(x), and f’’’(x). \(f(x)=\frac{x^2-7x}{x+1}\)

Use any method to evaluate the derivative of the following functions. \(f(x)=\frac{4-x^2}{x-2}\)

Use any method to evaluate the derivative of the following functions. \(f(x)=4x^2-\frac{2x}{5x+1}\)

Use any method to evaluate the derivative of the following functions. \(f(z)=z^2(e^{3z}+4)-\frac{2z}{z^2+1}\)

Use any method to evaluate the derivative of the following functions. \(h(r)=\frac{2-r-\sqrt{r}}{r+1}\)

Suppose the derivative of f exists, and assume that f(2) = 2 and f’(2) = 3. Let \(g(x)=x^2\cdot f(x)\) and \(h(x)=\frac{f(x)}{x-3}\). a. Find an equation of the line tangent to y = g(x) at x = 2. b. Find an equation of the line tangent to y = h(x) at x = 2.

The graph of \(y=\frac{a^3}{x^2+a^2}\), where a is a constant, is called the witch of Agnesi (named after the 18th-century Italian mathematician Maria Agnesi). a. Let a = 3 and find an equation of the line tangent to \(y=\frac{27}{x^2+9}\) and x = 2. b. Plot the function and the tangent line found in part (a).

Use the following table to find the given derivatives. \(\frac{d}{dx}\ [f(x)g(x)]\ |_{x=1}\)

Use the following table to find the given derivatives. \(\frac{d}{dx}\ [\frac{f(x)}{g(x)}]\ |_{x=2}\)

Use the following table to find the given derivatives. \(\frac{d}{dx}\ [x\ f(x)]\ |_{x=3}\)

Use the following table to find the given derivatives. \(\frac{d}{dx}\ [\frac{f(x)}{(x+2)}]\ |_{x=4}\)

Use the following table to find the given derivatives. \(\frac{d}{dx}\ [\frac{x\ f(x)}{g(x)}]\ |_{x=4}\)

Use the following table to find the given derivatives. \(\frac{d}{dx}\ [\frac{f(x)g(x)}{x}]\ |_{x=4}\)

Suppose the line tangent to the graph of f at x = 2 is y = 4x + 1 and suppose y = 3x - 2 is the line tangent to the graph of g at x = 2. Find an equation of the line tangent to the following curves at x = 2. a. \(y=f(x)\ g(x)\) b. \(y=\frac{f(x)}{g(x)}\)

The magnitude of the electrostatic force between two point charges Q and q of the same sign is given by \(F(x)=\frac{kQq}{x^2}\), where x is the distance between the charges and \(k=9 \times\ 10^9\ Nm^2/C^2\) is a physical constant (C stands for coulomb, the unit of charge; N stands for newton, the unit of force). a. Find the instantaneous rate of change of the force with respect to the distance between the charges. b. For two identical charges with Q = q = 1C, what is the instantaneous rate of change of the force at a separation of x = 0.001 m? c. Does the instantaneous rate of change of the force increase or decrease with the separation? Explain.

The magnitude of the gravitational force between two objects of mass M and m is given by \(F(x)=-\frac{GMm}{x^2}\), where x is the distance between the centers of mass of the objects and \(G=6.7\ \times\ 10^{-11}\ Nm^2/kg^2\) is the gravitational constant (N stands for newton, the unit of force; the negative sign indicates an attractive force). a. Find the instantaneous rate of change of the force with respect to the distance between the objects. b. For two identical objects of mass M = m = 0.1 kg, what is the instantaneous rate of change of the force at a separation of x = 0.01 m? c. Does the instantaneous rate of change of the force increase or decrease with the separation? Explain.

Special Product Rule In general, the derivative of a product is not the product of the derivatives. Find non constant functions f and g such that the derivative of f g and f’ g’.

Special Quotient Rule In general, the derivative of a quotient is not the quotient of the derivatives. Find non constant functions f and g such that the derivative of f/g equals f’/g’.

Means and tangents Suppose f is differentiable on an interval containing a and b, and let P(a, f(a)) and Q(b, f(b)) be distinct points on the graph of f. Let c be the x-coordinate of the point at which the tangent lines to the curve at P and Q intersect, assuming that the tangent lines are not parallel (see figure). a. If \(f(x)=x^2\), show that c = (a + b)/2, the arithmetic mean of a and b, for real numbers a and b. b. If \(f(x)=\sqrt{x}\), show that \(c=\sqrt{ab}\), the geometric mean of a and b, for a > 0 and b > 0. c. If f(x) = 1/x, show that c = 2ab/(a + b), the harmonic mean of a and b, for a > 0 and b > 0. d. Find an expression for c in terms of a and b for any (differentiable) function f whenever c exists.

Let F = f/g be the quotient of two functions that are differentiable at x. a. Use the definition of F’ to show that \(\frac{d}{dx}\ [\frac{f(x)}{g(x)}]=\lim _{h \rightarrow 0}\ \frac{f(x + h) g(x)-f(x) g(x + h)}{h\cdot g(x + h)\cdot g(x)}\) b. Now add -f(x)g(x) + f(x)g(x) (which equals 0) to the numerator in the preceding limit to obtain \(\lim _{h \rightarrow 0}\ \frac{f(x + h) g(x)-f(x)g(x)+f(x)g(x)-f(x)g(x+h)}{h\cdot g(x + h)\cdot g(x)}\) Use this limit to obtain the Quotient Rule. c. Explain why F’ = (f/g)’ exists, whenever \(g(x)\ \neq\ 0\).

Assuming the first and second derivatives of f and g exist, find a formula for \(\frac{d}{dx^2}\ [f(x)g(x)]\).

Proof by induction: derivative of \(e^{kx}\) for positive integers k Proof by induction is a method in which one begins by showing that a statement, which involves positive integers, is true for a particular value (usually k = 1). In the second step, the statement is assumed to be true for k = n, and the statement is proved for k = n + 1, which concludes the proof. a. Show that \(\frac{d}{dx}(e^{kx})=ke^{kx}\) for k = 1. b. Assume the rule is true for k = n (that is, assume \(\frac{d}{dx}(e^{nx})=ne^{nx}\), and this implies that the rule is true for k = n + 1. (Hint: write \(e^{(n+1)x}\) as the product of two functions, and use the Product Rule.)

Derivative of \(e^{kx}\) for negative integers k Use the Quotient Rule and Exercise 78 to show that \(\frac{d}{dx}(e^{kx})=ke^{kx}\) for negative integers k.

Quotient Rule for the second derivative Assuming the first and second derivatives of f and g exist, find a formula for \(\frac{d^2}{dx^2}\ [\frac{f(x)}{g(x)}]\).

Product Rule for three functions Assume that f, g, and h are differentiable at x. a. Use the Product Rule (twice) to find a formula for \(\frac{d}{dx}\ [f(x)g(x)h(x)].\) b. Use for formula in (a) to find \(\frac{d}{dx}\ [e^{2x}(x-1)(x+3)]\).

One of the Leibniz Rules One of several Leibniz Rules in calculus deals with higher-order derivatives of products. Let \((fg)^{(n)}\) denote the nth derivative of the product f g for \(n\ \geq\ 1\). a. Prove that \((fg)^{(2)}=gf''+2f'g'+fg''\). b. Prove that, in general, \((f g)^{(n)}=\sum_{k=0}^{n}\left(\begin{array}{l} n \\ k \end{array}\right) f^{(k)} g^{(n-k)}\) where \(\left(\begin{array}{l} n \\ k \end{array}\right)=\frac{n !}{k !(n-k) !}\) are the binomial coefficients. c. Compare the result of (b) to the expansion of \((a+b)^n\).