Worksheet 05 - Special Derivatives

Section 05

Instructions

This worksheet covers the differentiation of exponential and logarithmic functions, alone and combined with powers. For each problem, find the derivative and simplify your result. Difficulty levels are indicated by (x) = basic, (xx) = moderate, (xxx) = challenging, (xxxx) = advanced.

Problem Set 1: Exponential Functions

1. Basic Exponential Derivatives (x)

Find the derivative of each function:

  1. \(f(x) = e^{3x}\)

  2. \(f(x) = 5e^x\)

  3. \(f(x) = e^{-x}\)

  4. \(f(x) = 3^x\)

    Hint: Write \(3 = e^{\ln(3)}\), so \(3^x = e^{\ln(3) \cdot x}\).

2. Exponential Functions with Chain Rule (xx)

Find the derivative of each function:

  1. \(f(x) = e^{x^2}\)

  2. \(f(x) = e^{2x+1}\)

  3. \(f(x) = 2^{3x}\)

  4. \(f(x) = e^{-x^2/2}\)

Hint: For c), write \(2 = e^{\ln(2)}\) as in Problem 1 d).

3. Products and Quotients with Exponentials (xxx)

Find the derivative of each function:

  1. \(f(x) = x^2 \cdot e^x\)

  2. \(f(x) = \frac{e^x}{x}\)

  3. \(f(x) = x \cdot e^{-x}\)

  4. \(f(x) = \frac{e^{2x}}{e^x + 1}\)

Problem Set 2: Logarithmic Functions

4. Basic Logarithmic Derivatives (x)

Find the derivative of each function:

  1. \(f(x) = \ln(x)\)

  2. \(f(x) = 3\ln(x)\)

  3. \(f(x) = \log_{10}(x)\)

  4. \(f(x) = \log_2(x)\)

Hint: For c) and d), first use the change of base formula from Session 01-05 to write the logarithm with \(\ln\).

5. Logarithmic Functions with Chain Rule (xx)

Find the derivative of each function:

  1. \(f(x) = \ln(3x)\)

  2. \(f(x) = \ln(x^2 + 1)\)

  3. \(f(x) = \ln(\sqrt{x})\)

  4. \(f(x) = \log_3(2x + 5)\)

Hint: For d), use the change of base formula as in Problem 4.

6. Products and Quotients with Logarithms (xxx)

Find the derivative of each function:

  1. \(f(x) = x \cdot \ln(x)\)

  2. \(f(x) = \frac{\ln(x)}{x}\)

  3. \(f(x) = \ln(x)^2\)

  4. \(f(x) = x^2 \cdot \ln(x) - \frac{x^2}{2}\)

Problem Set 3: Powers, Exponentials and Logarithms Combined

7. Sums and Constant Factors (x)

Find the derivative of each function:

  1. \(f(x) = 2e^x - 5\ln(x)\)

  2. \(f(x) = x^4 + 6\ln(x)\)

  3. \(f(x) = \frac{e^x}{4} - \frac{3}{x}\)

  4. \(f(x) = \ln(x^3) + 7e^x\)

Hint: For d), first use the power law of logarithms from Session 01-05.

8. Chain Rule with Exponentials and Logarithms (xx)

Find the derivative of each function:

  1. \(f(x) = e^{5 - 2x}\)

  2. \(f(x) = (e^x + 3)^2\)

  3. \(f(x) = \ln(x^3 + 2x)\)

  4. \(f(x) = \sqrt{\ln(x)}\)

9. Products and Quotients with Powers (xxx)

Find the derivative of each function:

  1. \(f(x) = (x + 2) \cdot e^{3x}\)

  2. \(f(x) = \frac{x^2}{\ln(x)}\)

  3. \(f(x) = x^4 \cdot \ln(x)\)

  4. \(f(x) = \frac{e^x - 1}{e^x + 1}\)

Problem Set 4: Mixed Functions

10. Combining All Function Types (xxx)

Find the derivative of each function:

  1. \(f(x) = e^x \cdot \ln(x)\)

  2. \(f(x) = \ln(x) \cdot e^{-x}\)

  3. \(f(x) = e^{\sqrt{x}}\)

  4. \(f(x) = \ln(1 + x e^x)\)

11. Business Applications (xxxx)

  1. A company’s revenue follows \(R(t) = 500 \cdot e^{0.03t}\) thousand euros, where \(t\) is in years. Find the rate of change of revenue at \(t = 5\) years and interpret it.

  2. The demand for a product is modeled by \(D(p) = 1000 \cdot e^{-0.02p}\), where \(p\) is the price in euros. Find \(D'(p)\) and evaluate it at \(p = 50\). What does the sign tell you?

  3. A new product’s monthly sales are modeled by \(S(t) = 100 + 40\ln(t + 1) - 4t\) units, where \(t \geq 0\) is the number of months since the launch. Find \(S'(t)\) and determine after how many months the monthly sales are highest.

Problem Set 5: Challenge Problems

12. Advanced Derivatives (xxxx)

Find the derivative of each function:

  1. \(f(x) = \ln\left(\frac{x^2 + 1}{x^2 - 1}\right)\)

  2. \(f(x) = x^2 \cdot e^{x} \cdot \ln(x)\)

  3. \(f(x) = x^x\) (Hint: rewrite using \(e^{\ln(\cdot)}\))

  4. \(f(x) = \frac{\ln(x^2)}{e^{2x}}\)