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}


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?


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
Calculate the marginal revenue for the 10th unit.

Answer: 1,840

Problem 5.1.5: The demand function for a product is given below:
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

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


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}}}


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


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


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}


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}


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:


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]

External Resources

Khan Academy: Derivative of a Natural Log

Khan Academy: Worked Example



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