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: $f'(x) = 2\sec^2(2x)$

• 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: $f'(x) = \cot(x)$