10th Lecture

Derivative of Natural Log

The derivative of the natural log function is given by: $\frac{d}{dx} \ln(x) = \frac{1}{x}$

Constant Rule Example

If $f(x) = 5 \ln(x)$, then $f'(x) = 5 \cdot \frac{1}{x} = \frac{5}{x}$.

Product Rule Example

Given $f(x) = x^2 \ln(x)$, apply the product rule:

$\frac{d}{dx} (x^2 \ln(x)) = 2x \ln(x) + x^2 \cdot \frac{1}{x} = 2x \ln(x) + x = x(2\ln(x) + 1)$

Derivative of Log Base b

The derivative of a logarithm with base b is:

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

Conversion Formula

$\log_b(x) = \frac{\ln(x)}{\ln(b)}$

Example

$\frac{d}{dx} \log<em>2(x^4) = \frac{d}{dx} (4 \log</em>2(x)) = \frac{4}{x \ln(2)}$

Chain Rule with Natural Log

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

Example

$\frac{d}{dx} \ln(x^2 + 1) = \frac{2x}{x^2 + 1}$

$\frac{d}{dx} \log_2(x^3 + x) = \frac{3x^2 + 1}{(x^3 + x) \ln(2)}$

$\frac{d}{dx} \ln((x + 2)(x^2 + x)) = \frac{(x + 2)'(x^2 + x) + (x + 2)(x^2 + x)'}{(x + 2)(x^2 + x)} = \frac{(x^2 + x) + (x + 2)(2x + 1)}{(x + 2)(x^2 + x)}$

Derivative of Exponential Function

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

Example

$\frac{d}{dx} \frac{e^x}{x} = \frac{e^x \cdot x - e^x \cdot 1}{x^2} = \frac{e^x(x - 1)}{x^2}$

Derivative of b^x

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

Example

$\frac{d}{dx} 3^x = 3^x \ln(3)$

Chain Rule Example

$\frac{d}{dx} e^{x^2 + 1} = e^{x^2 + 1} \cdot (2x) = 2x \cdot e^{x^2 + 1}$

$\frac{d}{dx} 2^{3x} = 2^{3x} \ln(2) \cdot 3 = 3 \ln(2) \cdot 2^{3x}$

Application

Exponential growth model: $A(t) = A_0 \cdot b^t$

Where:

• $A_0$ is the initial quantity.

• $b$ is the growth factor.

• $t$ is time.

AIDS Epidemic Example

Given $A(t) = 1600 \cdot 2.25^t$, the rate of new cases per year is:

$A'(t) = 1600 \cdot 2.25^t \cdot \ln(2.25)$

For 1993 (t = 10):

$A'(10) = 1600 \cdot 2.25^{10} \cdot \ln(2.25) \approx 4,300,000$