DIFFERENTIATION 09

Exponential and Logarithmic Derivatives

Exponential and logarithmic functions have derivative rules that are closely connected through the number \(e\).

The main idea

The most important exponential derivative is

\[ \boxed{ \frac{d}{dx}e^x=e^x } \]

The natural exponential function is special because its derivative is itself.

1. Why \(e^x\) is special

Most functions change into a different expression when differentiated.

For example,

\[ x^3 \longrightarrow 3x^2. \]

But

\[ e^x \longrightarrow e^x. \]

Its instantaneous rate of change at every point equals its current value.

2. Example

\[ y=e^x, \]

then

\[ \frac{dy}{dx}=e^x. \]

At \(x=0\),

\[ \left. \frac{dy}{dx} \right|_{x=0} = e^0 = 1. \]

3. Exponential functions with an inner function

\[ y=e^{g(x)}, \]

use the chain rule:

\[ \boxed{ \frac{dy}{dx} = e^{g(x)}g'(x) } \]

For example,

\[ y=e^{3x}. \]

Then

\[ \frac{dy}{dx} = 3e^{3x}. \]

4. Another example

Consider

\[ y=e^{x^2}. \]

The outer derivative is

\[ e^{x^2}, \]

and the inner derivative is

\[ 2x. \]

Therefore,

\[ \boxed{ \frac{dy}{dx} = 2xe^{x^2} } \]

5. General exponential base

For a constant base \(a>0\),

\[ \boxed{ \frac{d}{dx}a^x = a^x\ln a } \]

The extra factor \(\ln a\) appears because \(e\) is the natural base for differentiation.

6. Why the formula works

Write

\[ a^x=e^{x\ln a}. \]

Differentiate using the chain rule:

\[ \frac{d}{dx}a^x = e^{x\ln a}\ln a. \]

Since

\[ e^{x\ln a}=a^x, \]

we obtain

\[ \frac{d}{dx}a^x = a^x\ln a. \]

7. The natural logarithm

The most important logarithmic derivative is

\[ \boxed{ \frac{d}{dx}\ln x = \frac{1}{x}, \qquad x>0. } \]

So the derivative of the natural logarithm is the reciprocal function.

8. Why this is connected to \(e^x\)

The functions

\[ e^x \]

and

\[ \ln x \]

are inverse functions.

Their derivative rules are therefore closely connected.

9. Logarithm with an inner function

\[ y=\ln(g(x)), \]

then by the chain rule,

\[ \boxed{ \frac{dy}{dx} = \frac{g'(x)}{g(x)} } \]

provided \(g(x)>0\).

10. Example

Let

\[ y=\ln(2x+1). \]

Then

\[ g(x)=2x+1 \]

and

\[ g'(x)=2. \]

Therefore,

\[ \boxed{ \frac{dy}{dx} = \frac{2}{2x+1} } \]

11. Example with a power

Consider

\[ y=\ln(x^2+1). \]

Then

\[ \frac{dy}{dx} = \frac{2x}{x^2+1}. \]

The numerator comes from differentiating the inside.

12. Logarithms with other bases

For \(a>0\) and \(a\neq1\),

\[ \log_a x = \frac{\ln x}{\ln a}. \]

Since \(\ln a\) is constant,

\[ \boxed{ \frac{d}{dx}\log_a x = \frac{1}{x\ln a} } \]

13. Exponential growth and decay

Consider

\[ P(t)=P_0e^{kt}. \]

Differentiate with respect to time:

\[ \frac{dP}{dt} = kP_0e^{kt}. \]

But

\[ P_0e^{kt}=P(t). \]

Therefore,

\[ \boxed{ \frac{dP}{dt}=kP } \]

The rate of change is proportional to the current value.

14. Logarithmic differentiation idea

Logarithms can also simplify differentiation when variables appear in powers or complicated products.

For example, if

\[ y=x^x, \]

take logarithms:

\[ \ln y=x\ln x. \]

Differentiating gives

\[ \frac{1}{y}\frac{dy}{dx} = \ln x+1. \]

Therefore,

\[ \boxed{ \frac{dy}{dx} = x^x(\ln x+1) } \]

15. The key rules

\[ \frac{d}{dx}e^x=e^x \] \[ \frac{d}{dx}e^{g(x)} = e^{g(x)}g'(x) \] \[ \frac{d}{dx}a^x = a^x\ln a \] \[ \frac{d}{dx}\ln x = \frac{1}{x} \] \[ \frac{d}{dx}\ln(g(x)) = \frac{g'(x)}{g(x)} \]

These rules appear constantly in calculus, differential equations and mathematical modelling.