Derivatives of Exponential and Logarithmic Functions

Introduction

Exponential and logarithmic functions play a crucial role in calculus because they describe growth, decay, and scaling processes found in science, economics, and engineering.
In this lecture, you’ll learn how to find derivatives of both exponential functions and logarithmic functions, and how they relate to the natural base $e$.

Derivatives of Exponential Functions

A. The Natural Exponential Function

The most important exponential function is:

$$f(x)=e^x$$

Its derivative is special because it’s the same as the original function:

$$f^\prime(x)=\frac{d}{dx}[e^x]=e^x$$

👉 The rate of change of $$e^x$$ equals its value — this is why it’s so useful in modeling natural growth.

B. Exponential Function with a Constant Base

For a general exponential function:

$$f(x)=a^x (a>0 , a\neq0) =e^{ln{a^{x}}}=e^{x \cdot ln(a)}$$

the derivative is:

$$f^\prime(x)=e^{x \cdot ln(a)} \cdot ln(a) =e^{ln(a)^x} \cdot ln(a) =a^{x} \cdot ln(a)$$

$$f^\prime(x)=\frac{d}{dx}[a^x]=a^{x} \cdot ln(a)$$

✅ Example:

$$\frac{d}{dx}[2^x]=2^xln(2)$$

C. Exponential Function with an Inner Function

When the exponent is more than just $x$, we use the Chain Rule.

$$f(x)=e^{u(x)}$$

then

$$f^\prime(x)=\frac{d}{dx}[e^{u(x)}]=e^{u(x)}\cdot u^\prime(x)$$

and

$$f(x)=a^{u(x)}$$

then

$$f^\prime(x)=\frac{d}{dx}[a^{u(x)}]=ln(a)a^{u(x)}\cdot u^\prime(x)$$

✅ Example:

$$\frac{d}{dx}[e^{3x}]=e^{3x}\cdot(3x)^\prime=3e^{3x}$$

Derivatives of Logarithmic Functions

A. Natural Logarithm ln⁡(x)

$$\frac{d}{dx}[ln(x)]=\frac{1}{x}$$​

B. Logarithm with a Different Base

For $$f(x)=log_{a}(x)$$

$$f^\prime(x)=\frac{d}{dx}=\frac{1}{xln(a)}$$​

✅ Example:

$$\frac{d}{dx}[log_{2}(x)]=\frac{1}{xln(2)}$$​

C. Logarithmic Function with an Inner Function

$$f(x)=ln(u(x))$$

then by the Chain Rule:

$$f^\prime(x)=\frac{d}{dx}[ln(u(x))]=\frac{1}{u(x)}\frac{du}{dx}=\frac{1}{u(x)}\cdot u^\prime(x)$$​

and

$$f(x)=log_{a}(u(x))$$

then by the Chain Rule:

$$f^\prime(x)=\frac{d}{dx}[log_{a}(u(x))]=\frac{1}{u(x)ln(a)}\frac{du}{dx}$$

✅ Example:

​$$\frac{d}{dx}[log_{3}(2x^2+5x+1)]=\frac{1}{(2x^2+5x+1)ln(3)}\cdot(2x^2+5x+1)^\prime=\frac{4x+5}{(2x^2+5x+1)ln(3)}$$

⚠️ Common Mistakes to Avoid

• Forgetting to multiply by the derivative of the inner function u′(x)u'(x)u′(x).

• Using $\log (x)$ instead of $\ln (x)$ without considering the base.

• Dropping $\ln (a)$ when differentiating $$a^{x}$$.