Differentiation, a critical tool in calculus, allows us to calculate the rate at which a function is changing at any given point. Advanced differentiation techniques are essential for tackling more complex functions and mathematical problems.

## Basic Rules of Differentiation

**Power Rule**

The derivative of $x^n$ is $nx^{n-1}$.**Example:**$f(x) = x^3$**Solution:**$f'(x) = 3x^2$

**Exponential Rule**

The derivative of $e^u$ is $\frac{du}{dx}e^u$.**Example:**$f(x) = e^{2x}$**Solution:**$f'(x) = 2e^{2x}$

**Logarithmic Rule**$\ln(u)$ is $\frac{du}{dx}\frac{1}{u}$.**Example:**$f(x) = \ln(5x)$**Solution:**$f'(x) = \frac{1}{x}$

**Trigonometric Rules**- The derivative of $\sin(ax)$ is $a\cos(ax)$.
**Example:**$f(x) = \sin(3x)$**Solution:**$f'(x) = 3\cos(3x)$ - The derivative of $\cos(ax)$ is $-a\sin(ax)$.
**Example:**$f(x) = \cos(4x)$**Solution:**$f'(x) = -4\sin(4x)$ - The derivative of $\tan(ax)$ is $a\sec^2(ax)$.
**Example:**$f(x) = \tan(2x)$**Solution:**$f'(x) = 2\sec^2(2x)$

- The derivative of $\sin(ax)$ is $a\cos(ax)$.

**Inverse Trigonometric Rule**

The derivative of $\tan^{-1}(ax)$ is $\frac{a}{1+(ax)^2}$.**Example:**$f(x) = \tan^{-1}(3x)$**Solution:**$f'(x) = \frac{3}{1+(3x)^2}$

## Differentiation of Algebraic Expressions

**Constant Multiple Rule**

The derivative of $kf(x)$ is $kf'(x)$.**Example:**$f(x) = 7\cos(x)$**Solution:**$f'(x) = -7\sin(x)$

**Sum and Difference Rule**

The derivative of $f(x) \pm g(x)$ is $f'(x) \pm g'(x)$.**Example:**$f(x) = e^{x} - x^2$**Solution:**$f'((x) = e^{x} - 2x$

**Differentiation of Composite Functions**

**Chain Rule**

If $h(x) = f(g(x))$, then $h'(x)$ is $f'(g(x)) \cdot g'(x)$.**Example:**$f(x = \ln(\sin(x))$**Solution:**$f'(x) = \cot(x)$

Written by: Dr Rahil Sachak-Patwa

LinkedIn

Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.