The Derivative

J. Zachary Klingensmith

5 The Derivative

5.1 The Derivative

Practice Problems

Problem 5.1.1: Find the derivative for the following functions. Additionally, find the slope of the function when x=2.

a) $f(x)=x^3+4x^2-3$
b) $f(x)=x^2+\frac{1}{x^2}$
c) $f(x)=7x^{1/5}-4x^{4/7}$
d) $y=10$
e) $f(x)=\frac{9}{x^{11}}$
f) $f(x)=(x^2+10x-\frac{2}{x})(x^2-4)$
g) $f(x)=\frac{4x+1}{x+2}$
h) $f(x)=10x^3(x^2-3)$
i) $f(x)=\frac{x^2-3x+10}{\sqrt{x}-10}$

Answers

a) $f'(x)=3x^2+8x, f'(2)=28$
b) $f'(x)=2x-\frac{2}{x^3}, f'(2)=\frac{15}{4}$
c) $f'(x)=\frac{7}{5x^{4/5}}-\frac{16}{7x^{3/7}}, f'(2)=-0.89$
d) $y'=0, y'(2)=0$
e) $f'(x)=-\frac{99}{x^12}, f'(2)=-\frac{99}{4096}$
f) $f'(x)=\left(2x+10+2x^{-2}\right)\left(x^2-4\right)+\left(x^2+10x-2x^{-1}\right)\left(2x\right), f'(2)=92$
g) $f'(x)=\frac{7}{\left(x+2\right)^2}, f'(2)=\frac{7}{16}$
h) $f'(x)=50x^4-90x^2, f'(2)=440$

Problem 5.1.2: For what value(s) of x does the function below have a horizontal tangent line?

$f(x)=2x^3-3x^2-36x-1$

Answer: x=-2, 3

Problem 5.1.3: Determine the coefficients of a and b that make the statement below true:

$p(x)=x^2+ax+b, p(1)=0, p'(1)=7$

Answer: a=5, b=-6

Problem 5.1.4: A company finds that its revenue function is
$R(Q)=1800Q+32Q^2-2Q^3.$
Calculate the marginal revenue for the 10th unit.

Answer: 1,840

Problem 5.1.5: The demand function for a product is given below:
$P=500-Q.$
a) Calculate the marginal revenue when the firm sells its 20th unit.
b) At what level of production does marginal revenue equal 0?

Answer: 460, 250

Problem 5.1.6: Consider the function h(x) for which h(3)=6 and h'(3)=-5. Find f'(3) for the function
$f(x)=\frac{h(x)}{x}.$

Answer: -7/3

Problem 5.1.7: Find the derivative for the following functions. Additionally, find the slope of the function when x=2.

a) $y=6(x^2-4x)^{-3}$
b) $f(x)=\sqrt{6x+5}$
c) $g(x)=\frac{(2x+5)^4}{x-5}$
d) $h(x)=(x^2-3x+1)(3x-1)^3$

Answers

a) $y'=\frac{-36x+72}{\left(x^2-4x\right)^4}, y'(2)=0$
b) $f'(x)=\frac{3}{\left(6x+5\right)^{1/2}}, f'(2)=0.73$
c) $g'(x)=\frac{(8x-40)(2x+5)^{3}-(2x+5)^{4}}{(x-5)^{2}}, g'(2)=-2,673$
d) $h'(x)=(2x-3)(3x-1)^3+(9x^2-27x+9)(3x-1)^{2}, h'(2)=-100$

External Resources

Khan Academy: Power Rule

Khan Academy: Differentiating Integer Powers

Khan Academy: Product Rule (Ignore the Trig)

Khan Academy: Quotient Rule (Ignore the Trig)

Khan Academy: Chain Rule

Khan Academy: Worked Chain Rule Example

5.2 Exponentials and logarithms: A Review

Practice Problems

Problem 5.2.1: You place 1,000USD in an account which pays 6% compounded continuously. Find the future value of the account after 5 years? 10 years? 20 years? What if the account paid 8%?

Answers: 1,350, 1,822, 3,320; 1,492, 2,226, 4,953

Problem 5.2.2: After a child’s birth, an account is opened, and a single payment has been made, so what when the child is 18, they will receive 25,000USD. What initial deposit was made assuming the account earned 4% per year compounded continuously.

Answer: 12,169

Problem 5.2.3: The real GDP per capita of a country in a given year is 3,500USD. How long will it take for the GDP per capita to reach 10,000USD if the economy grows at a rate of 2% compounded continuously.

Answer: 52.5 years

Problem 5.2.4: Mimi deposits 10,000USD into an account which compounds continuously. After 30 years, the deposit is worth 50,000USD. What interest rate did this account pay?

Answer: 5.4%

Problem 5.2.5: Fully expand the following logarithms:

a) $\ln\left[{\frac{(x-1)^3x^2}{(x+2)^2}}\right]$
b) $\ln{\sqrt{\frac{x+1}{x-1}}}$

Answers:

a) $3\ln(x-1)+2\ln(x)-2\ln(x+2)$
b) $\frac{1}{2}\ln(x+1)-\frac{1}{2}\ln(x-1)$

Problem 5.2.6: Solve the following for x:

a) $4e^{2x-1}-1=5$
b) $3+4\ln{x}=15$
c) $e^{0.09t}=31$
d) $\ln{\ln{x}}=3$

Answers:

a) $x=\frac{\ln(3/2)+1}{2}\approx 0.703$
b) $x=e^{3}\approx 20.09$
c) $t=\frac{\ln(31)}{.09}\approx 38.2$
d) $x=e^e^3\approx 528,491,312$

External Resources

Khan Academy: e and Compound Interest

Khan Academy: Evaluating ln with a Calculator

5.3 The Derivative of implicit Equations

Practice Problems

Problem 5.3.1: Find dy/dx for the following:

a) $y^3+2y^2+3y-2x^2+3x-1=0$
b) $\sqrt{x+y}=x$
c) $x^2y+y^2x=1$
d) $\frac{x-1}{y^2+1}=2$

Answers:

a) $\frac{dy}{dx}=\frac{4x-3}{3y^2+4y+3}$

b) $\frac{dy}{dx}=2x-1$

c) $\frac{dy}{dx}=\frac{-2xy-y^2}{x^2+2xy}$

d) $\frac{dy}{dx}=\frac{1}{4y}$

External Resources

Khan Academy: Implicit Differentiation

5.4 The Derivatives of exponential and logarithmic functions

Practice Problems

Problem 5.4.1: Find the derivative of the following functions:

a) $f(x)=14e^x+x^{14}$
b) $g(x)=e^x+x^e$
c) $h(x)=e^{\sqrt{x}}$
d) $j(x)=4x^3e^{x^2}$

Answers:

a) $f'(x)=14e^x+14x^{13}$
b) $g'(x)=e^x+ex^{e-1}$
c) $h'(x)=\frac{e^{x^{1/2}}}{2x^{1/2}}$
d) $j'(x)=e^{x^2}\left(12x^2+8x^4\right)$

Problem 5.4.2: Find the derivative of the following functions:

a) $f(x)=(\ln{12})e^{3x}$
b) $g(x)=\ln \left[\frac{8-x}{8+x}}\right]$
c) $h(x)=\ln{\sqrt{x^2-4}}$
d) $j(x)=(\ln{x})^4$
e) $y=\frac{1+e^x}{1-e^x}$

Answers:

a) $f'(x)=3\ln(12)e^{3x}$
b) $g'(x)=\frac{-1}{8-x}-\frac{1}{8+x}$
c) $h'(x)=\frac{x}{x^2-4}$
d) $j'(x)=\frac{4}{x}\left(\ln(x)\right)^3$
e) $y'=\frac{2e^x}{\left(1-e^x)^2}$

Problem 5.4.3: Find the derivative of the following:

$y=\frac{(x^2-4x)^5(x^3+2x)^3(x-5)^2}{(x^3-5x^2)^4\sqrt{x^4-5x}}$

Answer: $\frac{dy}{dx}=\left[\frac{(x^2-4x)^5(x^3+2x)^3(x-5)^2}{(x^3-5x^2)^4\sqrt{x^4-5x}}\right]\left[\frac{10x-20}{x^2-4x}+\frac{9x^2+6}{x^3+2x}+\frac{2}{x-5}-\frac{12x^2-20}{x^3-5x}-\frac{4x^3-5}{2x^4-10x}\right]$