Many mathematicians have developed various rules for differentiation with which we can find the derivative of any given function. The more commonly used differential formulas are listed below:

**Derivative of a constant:** If a is a constant, then $\frac{da}{dx}$ = 0 or f ‘(a) = 0. This means that the derivative of any number or a constant will always be 0.

**n**^{th} power derivative: $\frac{d(x^{n})}{dx}$ or f‘(x^{n}) = nx^{n - 1}. This means the derivative of a variable which has some constant power is equal to the constant times its variable, provided the power of the variable decreases by 1.

For example, consider the derivatives of the following:

f‘(x^{3}) = 3x^{3 - 1} = 3x^{2}

f‘(x^{-3}) = -3x^{- 3 - 1} = -3x^{- 4}

**Multiplication by a constant:** If a is a constant, then $\frac{d(af(x))}{dx}$ = a$\frac{d(f(x))}{dx}$, or f’(af(x)) = af’(f(x)). This means that, if any constant term is multiplied by a function, then it will go outside the derivative of a function.

**Derivative of a single variable of power one:** $\frac{d}{dx}$$(x) = 1$. This means that the derivative of a variable having power as one will always be equal to 1. For example, consider f(x) = 2x

f'(x) = 2

**Addition Rule:** $\frac{d}{dx}$$[f(x)+g(x)]$ = $\frac{d}{dx}$$f(x)+$$\frac{d}{dx}$$g(x)$, which means the derivative of a sum is the sum of the individual derivatives.

**Subtraction Rule:** Just like the addition rule, the subtraction rule says, that the derivative of the difference of two functions will be the difference of their individual derivatives, i.e. $\frac{d}{dx}$$[f(x)-g(x)]$ = $\frac{d}{dx}$$f(x)-$$\frac{d}{dx}$$g(x)$

**Product Rule:** The product rule says that the derivative of the product of two functions will be equal to the second function multiplied with the derivative of the first function plus the first function multiplied with the derivative of the second function, i.e.

$\frac{d}{dx}$$(f(x)g(x)) = f(x)g'(x) + g(x)f'(x)$

**Quotient Rule: **The quotient rule says that the derivative of the division of two functions, that is first function divided by the second function, will be equal to the second function multiplied with the derivative of the first function minus the first function multiplied with the derivative of the second function (this will be the numerator), this whole expression divided by the square of the second function, which will be the denominator, i.e.

$\frac{d}{dx}\frac{f(x)}{g(x)} = \frac{g(x)f'(x)- f(x)g'(x)}{(g(x))^{2}}$

**Derivative of log function:** $\frac{d}{dx}$$log y$ = $\frac{1}{y} \frac{dy}{dx}$

**Derivative of exponent function: **$\frac{d}{dx}$$e^{y} = e^{y}$$\frac{dy}{dx}$

**Chain Rule:** If y = f (g (x)), that is, y is a function of f which is in itself a function of g, then the chain rule is defined as, y’(x) = f’ (g(x)) x g’(x), where g’(x) represents the derivative of function g(x). Chain rule can be summarized as finding the derivative of the outer function first and then, multiplying it with the derivative of the inner function.

The following is the list of some of the trigonometric derivatives formulas:

$\frac{d}{dx}$ sin x = cos x, $\frac{d}{dx}$ cosx = -sinx

$\frac{d}{dx}$ (tan x) = sec^{2}x, $\frac{d}{dx}$ (cot x) = -csc^{2}x

$\frac{d}{dx}$ (sec x) = sec x tan x, $\frac{d}{dx}$ (csc x) = -csc x cot x