?Extended Product Rule The Product Rule can be extended to the product of three

Find the derivative of \(f(x)=\left(1+2 x^{2}\right)\left(x-x^{2}\right)\) in two ways: by using the Product Rule and by performing the multiplication first. Do your answers agree? Equation Transcription: Text Transcription: f(x)=(1+2x^2)(x-x^2)

Find the derivative of the function \(F(x)=\frac{x^{4}-5 x^{3}+\sqrt{x}}{x^{2}}\) in two ways: by using the Quotient Rule and by simplifying first. Show that your answers are equivalent. Which method do you prefer? Equation Transcription: Text Transcription: F(x)=x^4 - 5x^3 + sqrt x / x^2

Differentiate. \(y=\left(4 x^{2}+3\right)(2 x+5)\)

Differentiate \(y=\left(10 x^{2}+7 x-2\right)\left(2-x^{2}\right)\)

Differentiate \(y=x^{3} e^{x}\) Equation Transcription: Text Transcription: y=x^3 e^x

Differentiate \(y=\left(e^{x}+2\right)\left(2 e^{x}-1\right)\) Equation Transcription: Text Transcription: y=(e^x+2)(2e^x-1)

Differentiate \(f(x)=\left(3 x^{2}-5 x\right) e^{x}\) Equation Transcription: Text Transcription: f(x)=(3x^2-5x)e^x

Differentiate \(g(x)=(x+2 \sqrt{x}) e^{x}\) Equation Transcription: Text Transcription: g(x)=(x+2 sqrt x) e^x

Differentiate \(y=\frac{x}{e^{x}}\) Equation Transcription: Text Transcription: y=x/e^x

Differentiate \(y=\frac{e^{x}}{1-e^{x}}\) Equation Transcription: Text Transcription: y=e^x / 1 - e^x

Differentiate \(g(t)=\frac{3-2 t}{5 t+1}\) Equation Transcription: Text Transcription: g(t)=3 - 2t / 5t - 1

Differentiate \(G(u)=\frac{6 u^{4}-5 u}{u+1}\) Equation Transcription: Text Transcription: G(u)=6u^4 - 5u / u + 1

Differentiate. \(\mathrm{f}(\mathrm{t})=\frac{5 t}{t^{3}-t-1}\)

Differentiate. \(F(x)=\frac{1}{2 x^{3}-6 x^{2}+5}\)

Differentiate. \(y=\frac{s-\sqrt{s}}{s^{2}}\) Equation Transcription: Text Transcription: y = s - sqrt s / s^2

Differentiate. \(\mathrm{y}=\frac{\sqrt{x}}{\sqrt{x}+1}\)

Differentiate. \(J(u)=\left(\frac{1}{u}+\frac{1}{u^{2}}\right)\left(u+\frac{1}{u}\right)\) Equation Transcription: Text Transcription: J(u) = (1/u + 1/u^2)(u + 1/u)

Differentiate. \(h(w)=\left(w^{2}+3 w\right)\left(w^{-1}-w^{-4}\right)\)

Differentiate. \(H(u)=(u-\sqrt{u})(u+\sqrt{u})\) Equation Transcription: Text Transcription: H(u) = (u - sqrt u)(u + sqrt u)

Differentiate. \(f(z)=\left(1-e^{z}\right)\left(z+e^{z}\right)\) Equation Transcription: Text Transcription: f(z) = (1 - e^z)(z + e^z)

Differentiate. \(V(t)=\left(t+2 e^{t}\right) \sqrt{t}\)

Differentiate. \(W(t)=e^{t}\left(1+t e^{t}\right)\) Equation Transcription: Text Transcription: W(t) = e^t (1 + te^t)

Differentiate. \(y=e^{p}(p+p \sqrt{p})\) Equation Transcription: Text Transcription: y = e^P (p + p sqrt p)

Differentiate. \(h(r)=\frac{a e^{r}}{b+e^{r}}\)

Differentiate. \(f(t)=\frac{\sqrt[3]{t}}{t-3}\) Equation Transcription: Text Transcription: f(t) = cuberoot t / t - 3

Differentiate.

Differentiate.

Differentiate.

Differentiate. \(f(x)=\frac{x}{x+\frac{c}{x}}\)

Differentiate. \(f(x)=\frac{a x+b}{c x+d}\)

Find and .

Find and

Find \(f^{\prime}(x)\) and \(f^{\prime \prime}(x)\). \(f(x)=\frac{x}{x^{2}-1}\) Equation Transcription: Text Transcription: f' (x) f'' (x) f(x) = x / x^2 - 1

Find \(f^{\prime}(x)\) and \(f^{\prime \prime}(x)\). \(f(x)=\frac{x}{1+\sqrt{x}}\) Equation Transcription: Text Transcription: f(x) = x / 1 + sqrt x Image text transcription: Find f 9sxd and f 99sxd. f sxd ? x 1 1 sx

Find an equation of the tangent line to the given curve at the specified point. \(y=\frac{x^2}{1+x},\quad\ \ \left(1,\frac{1}{2}\right)\) Equation Transcription: Text Transcription: y = x^2 / 1 + x, (1, 1/2)

Find an equation of the tangent line to the given curve at the specified point. \(y=\frac{1+x}{1+e^x}, \quad\left(0, \frac{1}{2}\right)\)

Find equations of the tangent line and normal line to the given curve at the specified point. \(y=\frac{3 x}{1+5 x^{2}},\left(1, \frac{1}{2}\right)\)

Find equations of the tangent line and normal line to the given curve at the specified point. ,

(a) The curve is called a witch of Maria Agnesi. Find an equation of the tangent line to this curve at the point . (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.

(a) The curve is called a serpentine. Find an equation of the tangent line to this curve at the point (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.

(a) If , find . (b) Check to see that your answer to part (a) is reasonable by comparing the graphs of and .

(a) If , find and . (b) Check to see that your answers to part (a) are reasonable by comparing the graphs of , and .

If \(f(x)=x^{2} /(1+x)\), find \(f^{\prime\prime}\ (1)\).

If , find .

Suppose that , and Find the following values. (a) (b) (c)

Suppose that f(4) = 2, g(4) = 5, \(f^{\prime}(4)=6\), and \(g^{\prime}(4)=-3\). Find \(h^{\prime}(4)\). (a) h(x) = 3f(x) + 8g(x) (b) h(x) = f(x) g(x) (c) \(h(x)=\frac{f(x)}{g(x)}\) (d) \(h(x)=\frac{g(x)}{f(x)+g(x)}\)

If , where and , find .

If h(2) = 4 and \(h^{\prime}(2)=-3\), find \(\left.\frac{d}{d x}\left(\frac{h(x)}{x}\right)\right |_{x=2}\)

If , where and , find an equation of the tangent line to the graph of at the point where .

If and for all , find .

If and are the functions whose graphs are shown, let and (a) Find . (b) Find .

Let P(x)=F(x) G(x) and Q(x)=F(x)/G(x), where F and G are the functions whose graphs are shown. (a) Find \(P^{\prime}(2)\). (b) Find \(Q^{\prime}(7)\).

If g is a differentiable function, find an expression for the derivative of each of the following functions. (a) y=x g(x) (b) \(y=\frac{x}{g(x)}\) (c) \(y=\frac{g(x)}{x}\)

If is a differentiable function, find an expression for the derivative of each of the following functions. (a) (b) (c) (d)

How many tangent lines to the curve pass through the point At which points do these tangent lines touch the curve?

Find equations of the tangent lines to the curve that are parallel to the line .

Find , where Hint: Instead of finding first, let be the numerator and be the denominator of , and compute from , and .

Use the method of Exercise 57 to compute , where

In this exercise we estimate the rate at which the total personal income is rising in Boulder, Colorado. In 2015, the population of this city was 107,350 and the population was increasing by roughly 1960 people per year. The average annual income was per capita, and this average was increasing at about $2250 per year (a little above the national average of about $1810 yearly). Use the Product Rule and these figures to estimate the rate at which total personal income was rising in Boulder in 2015. Explain the meaning of each term in the Product Rule.

A manufacturer produces bolts of a fabric with a fixed width. The quantity of this fabric (measured in yards) that is sold is a function of the selling price (in dollars per yard), so we can write . Then the total revenue earned with selling price is . (a) What does it mean to say that and (b) Assuming the values in part (a), find and interpret your answer.

The Michaelis-Menten equation for the enzyme chymotrypsin is \(v=\frac{0.14[\mathrm{S}]}{0.015+[\mathrm{S}]}\) where v is the rate of an enzymatic reaction and [S] is the concentration of a substrate S. Calculate dv / d[S] and interpret it.

The biomass of a fish population is the total mass of the members of the population at time . It is the product of the number of individuals in the population and the average mass of a fish at time . In the case of guppies, breeding occurs continually. Suppose that at time weeks the population is 820 guppies and is growing at a rate of 50 guppies per week, while the average mass is and is increasing at a rate of week. At what rate is the biomass increasing when

Extended Product Rule The Product Rule can be extended to the product of three functions. (a) Use the Product Rule twice to prove that if , and are differentiable, then (b) Taking in part (a), show that (c) Use part (b) to differentiate .

(a) If , where and have derivatives of all orders, show that (b) Find similar formulas for and . (c) Guess a formula for .

Find expressions for the first five derivatives of . Do you see a pattern in these expressions? Guess a formula for and prove it using mathematical induction.

Reciprocal Rule If is differentiable, the Reciprocal Rule says that (a) Use the Quotient Rule to prove the Reciprocal Rule. (b) Use the Reciprocal Rule to differentiate the function in Exercise (c) Use the Reciprocal Rule to verify that the Power Rule is valid for negative integers, that is, for all positive integers .