Product rule and Quotient rule in derivative

In this post  we are going to  tell you about product rule and quotient rule in derivative.

Product rule- The product rule is a  rule for differentiating problems when one function is multiplied by another. The rule  is 

\[\frac{d}{dx}(f.g)=f\frac{d}{dx}(g)+g\frac{d}{dx}(f)\]

In above rule we can see that \(f(x)\) and \(g(x)\) are multiplied with each other . Lets try an example

Find the derivative of \[(x+4)^4 . \sin x\]
in the above example we are taking \(f(x)=(x+4)^4\) and \(g(x)=\sin x\) ,now we are going to apply rule \[\frac{d}{dx}(f.g)=f\frac{d}{dx}(g)+g\frac{d}{dx}(f)\]
 
here we use the derivative formulas you can check here . so the answer \[ (x+4)^4 .\cos x + 4\sin x . (x+4)^3\]

Quotient rule- The quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. i.e \(\displaystyle \frac{f(x)}{g(x)}\) .The rule to differentiate the given function is \[\frac{d}{dx}\left ( \frac{f}{g} \right )=\frac{g(x).f'(x) - f(x)g'(x)}{[g(x)]^2}\] here \(f'(x)\) is derivative of \(f(x)\) and \(g'(x)\) is derivative of \(g(x)\)

Lets try an example- find the derivative of \(\displaystyle \frac{x^2}{(x+1)}\)
while solving the question we will take \(f(x)=x^2\) and \(g(x)=(x+1)\)

by the quotient rule \[\frac{d}{dx}\left ( \frac{x^2}{x+1} \right )=\frac{(x+1).2x - x^2}{[(x+1)]^2}\] \[=\frac{2x^2+2x- x^2.1}{(x+1)^2}\] \[=\frac{x^2+2x}{(x+1)^2}\] Now try to solve some examples
\(1)  \sin x . \cos x \)
\(2)   x^3 .(x+10)^8\)
\(3)   \frac{x+1}{x^3}\)
\(4)   (x+1)^4.\sin x\)
\(5)  \sin x .\cos x. (x+1)\)