# Differentiation of Functions

## Differentiation of Functions

# Differentiation of Exponential Functions

**Exponential functions**describe phenomena that change at rates proportional to the current value such as population growth.- If you’re differentiating an exponential function in the form
**e^x**, the derivative is simply**e^x**. - For differentiation of the form
**a^x**where a > 0 and a ≠ 1, we must know that the derivative is**a^x * ln(a)**. - When differentiating the function
**e^(f(x))**, we apply the**chain rule**, resulting in**f’(x)e^(f(x))**.

# Differentiation of Logarithmic Functions

**Logarithmic differentiation**is very useful for functions of the form**f(x) = [g(x)]^h(x)**.- The derivative of the
**natural logarithm**function is**1/x**. - The
**Chain Rule**applies here too. The derivative of ln(f(x)) is**f’(x)/f(x)**. - For
**other bases apart from e**(e.g. 10 or 2), use the change of base formula to convert to natural logarithms first.

# Differentiation of Trigonometric Functions

- Familiarity with derivatives of the
**basic trigonometric functions**such as sine, cosine and tangent is necessary. - The derivative of
**sin(x)**is**cos(x)**, and the derivative of**cos(x)**is**-sin(x)**. - The derivative of
**tan(x)**is**sec^2(x)**. Remember that sec(x) = 1/cos(x). - For
**other trigonometric functions**such as cot(x), sec(x), and csc(x), the key is to express these functions in terms of sin(x) and cos(x) and then apply the quotient rule or other basic differentiation rules. - The derivatives of invasive trigonometric functions (arcsin, arccos, arctan) can also be obtained by using implicit differentiation and pythagorean identities.

# Differentiation of Hyperbolic Functions

- The
**hyperbolic functions**sinh(x), cosh(x) and tanh(x) can also be differentiated. - The derivative of
**sinh(x)**is**cosh(x)**, and the derivative of**cosh(x)**is**sinh(x)**. - The derivative of
**tanh(x)**is**sech^2(x)**, where sech(x) = 1/cosh(x). - Derivatives of other hyperbolic functions like coth(x), sech(x) and csch(x) can be found by expressing them in terms of sinh(x) and cosh(x) and using other rules of differentiation.