Derivative Rules

Mathematically, the derivatives are used to find the slope of a line or slope of a curve. Derivative rules are used to find the derivatives of different operations and different types of functions such as Trigonometric functions, power functions, logarithmic functions, exponential functions, etc. The following are some of the important derivative rules   

Sum Rule 

ddx(f(x)+g(x))     =   f'(x)+g'(x)

Difference Rule

ddx(f(x)−g(x))   =  f'(x)−g'(x).

Power Rule

Let n$\mathrm{ n}$ be a positive integer. If f(x)=xn$$,then

f'(x)=nxn−1.

Chain Rule 

If y is function u and u is a function x , then

dydx=dydu⋅dudx.

Product Rule

ddx(f(x)g(x))   =  f'(x)g(x)+g'(x)f(x).

Quotinet Rule

 ddx(f(x)g(x))=ddx(f(x))⋅g(x)−ddx(g(x))⋅f(x)(g(x))2.                                 

                        =   

f'(x)g(x)−g'(x)f(x)(g(x))2.(Ai

Partial Derivatives 

Suppose, we have a function f(x, y), which depends on two variables x and y, where x and y are independent of each other. Then we say that the function f partially depends on x and y.

So, the partial derivative of f with respect to x will be ∂f/∂x keeping y as constant. Similarly the partial derivative of f with respect to y is ∂f/∂y keeping x constant.

fx = ∂f / ∂xfy = ∂f / ∂y second order derivative of x is given by fxx = ∂²f / ∂x² = ∂ / ∂x (∂f / ∂x) = ∂ / ∂x (fx)Second order derivative of  y is given by    fyy = ∂²f / ∂y² = ∂ / ∂y (∂f / ∂y) = ∂ / ∂x (fy)Similary the second order derivative of xy or yx is given byfxy = ∂²f / ∂y ∂x = ∂ / ∂y (∂f / ∂x) = ∂ / ∂y (fx) fyx = ∂²f / ∂x ∂y = ∂ / ∂x (∂f / ∂y) = ∂ / ∂x (fy)If y = f(x) is a function where x is again a function of two variables u and v (i.e., x = x (u, v)) then∂f/∂u = ∂f/∂x · ∂x/∂u;∂f/∂v = ∂f/∂x · ∂x/∂vThe chain rule ,quotient rule and partial derivatives  are extensively used in AI (Artificial Intelligence), particularly the chain rule for computing the weights using back propagation.