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)).

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.

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.

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.