Lecture: Fundamental Derivative Rules — Power, Exponential, Logarithmic, Sum & Difference

Objective

To learn and apply key rules of differentiation that allow us to compute derivatives quickly and reliably without using the limit definition every time.

Review: Why Rules?

In the last lecture, we used the limit definition to compute the derivative of functions like:

• $f(x) = x^2 \Rightarrow f'(x) = 2x$
• $f(x) = \frac{1}{x} \Rightarrow f'(x) = -\frac{1}{x^2}$

These calculations can be tedious. Fortunately, mathematics gives us general rules that apply to broad classes of functions — making computation efficient and systematic.

The Power Rule

If $f(x) = x^n$, then:

$$f'(x) = nx^{n - 1}, \quad \text{for any real number } n$$

Examples:

• $f(x) = x^3 \Rightarrow f'(x) = 3x^2$
• $f(x) = x^{-2} \Rightarrow f'(x) = -2x^{-3}$
• $f(x) = \sqrt{x} = x^{1/2} \Rightarrow f'(x) = \frac{1}{2}x^{-1/2}$

Exponential Functions: $e^x$

The natural exponential function has a beautiful property:

$$\frac{d}{dx}(e^x) = e^x$$

Natural Logarithm: $\ln(x)$

For $x > 0$:

$$\frac{d}{dx}(\ln x) = \frac{1}{x}$$

This was proven in the last lecture using the limit definition.

Sum and Difference Rule

If $f(x) = g(x) + h(x)$, then:

$$f'(x) = g'(x) + h'(x)$$

If $f(x) = g(x) - h(x)$, then:

$$f'(x) = g'(x) - h'(x)$$

Example:

$$f(x) = x^3 + 5x^2 - 7x + 9 \Rightarrow f'(x) = 3x^2 + 10x - 7$$

Constant Multiple Rule

If $f(x) = c \cdot g(x)$, then:

$$f'(x) = c \cdot g'(x)$$

Example:

$$f(x) = 5x^3 \Rightarrow f'(x) = 5 \cdot 3x^2 = 15x^2$$

Summary Table of Rules

Practice Exercises

1. Apply Power Rule

Differentiate:

• (a) $f(x) = x^5$
• (b) $f(x) = x^{-3}$
• (c) $f(x) = \sqrt[3]{x}$
• (d) $f(x) = \frac{1}{x^4}$

2. Sum/Difference and Constant Multiples

Differentiate:

• (a) $f(x) = 3x^4 + 5x^2 - 7$
• (b) $f(x) = x^3 - \frac{1}{x} + 4$
• (c) $f(x) = 2\sqrt{x} + 5x^{1/3}$

3. Exponential and Logarithmic Derivatives

Simplify the expression using log identities before differentiating:

• (a) $f(x) = 5e^x$
• (b) $f(x) = \ln(x^3)$
• (c) $f(x) = \ln\left(\frac{x^2}{\sqrt{x}}\right)$
• (d) $f(x) = \ln(e^{x^3})$

4. Mix It Up

Differentiate:

• (a) $f(x) = e^x + x^2 - \frac{1}{x}$
• (b) $f(x) = x^3 + \ln x$

5. Rewriting Functions to Apply Rules

Differentiate the following functions by rewriting them into a form where the power rule or other known rules can be applied.

• (a) $f(x) = \frac{\sqrt[4]{x^3}}{x^2}$
• (b) $f(x) = \frac{5}{x^{1/3}} + 3\sqrt{x} - \frac{1}{x^2}$
• (c) $f(x) = \frac{x^2 + \sqrt{x}}{x^{1/2}}$
• (d) $f(x) = \frac{1}{\sqrt[3]{x^2}} + \frac{2}{x^{1/5}}$

Extra Advanced Problems

1. Function with Parameters (for analysis)

Let:

$$f(x) = ax^n + b\ln(x) + ce^x$$

• (a) Compute $f'(x)$ in terms of $a, b, c, n$
• (b) For what values of $a, b, c$ is $f'(x) = 0$ at $x = 1$?
• (c) Analyze whether $f(x)$ can have a local minimum at $x = 1$

2. Piecewise Polynomial Analysis

Let

$$f(x) = \begin{cases} x^3 + 2x^2 - x + 1 & \text{if } x < 1 \\ 4x^2 - 3x + 2 & \text{if } x \geq 1 \end{cases}$$

Tasks:

• (a) Is $f(x)$ continuous at $x = 1$?
• (b) Is $f(x)$ differentiable at $x = 1$?

Use left-hand and right-hand derivatives.

3. Log-Exponential Combination

Let

$$f(x) = \ln x + 5e^x - 3x^3 + 2x$$

Tasks:

• (a) Find $f'(x)$
• (b) Find all $x > 0$ such that $f'(x) = 0$

Use algebra to simplify, and solve graphically or numerically if needed.

4. Prove the Power Rule

Use the limit definition to prove the power rule for:

$f(x) = x^n$, where $n \in \mathbb{N}$

5. Cost-Analysis Word Problem

A company’s total cost to produce $x$ units is modeled by:

$$C(x) = 10x^3 - 15x^2 + 500x + 1000$$

Tasks:

• (a) Find the marginal cost function $C'(x)$
• (b) Compute $C'(10)$. Interpret the result.
• (c) For which $x$ is the marginal cost minimized?

6. Population Dynamics

The population of bacteria over time $t$ (in hours) is:

$$P(t) = -100e^t + 50t + 18000$$

Tasks:

• (a) Find $P'(t)$
• (b) Find when the population stops increasing
• (c) Find when the population seizes to exist

7. Logarithmic Inverse

• (a) Show that $f(x) = \ln(x)$ is the inverse of $g(x) = e^x$, and that:

$$\frac{d}{dx} \ln(x) = \frac{1}{e^{\ln(x)}} = \frac{1}{x}$$

Wrap-Up

You now know the core rules of differentiation:

• Power Rule
• Logarithmic Rule
• Exponential Rule
• Sum, Difference, and Constant Multiple Rules

These rules are the foundation of fast differentiation and will prepare you for the next tools: product, quotient, and chain rules.