Question

I am having difficulty solving derivatives. I was wondering if you could explain the product and quotient rule with examples? Thank you in advance!

Answer 1

Given:

Product and quotient rule.

Required:

Explain the product and quotient rule with examples.

Explanation:

Product rule:

If two functions are in the multiple forms then the derivative of two functions is given by the formula:

$\frac{d}{dx}(u.v)=v\frac{d}{dx}u+u\frac{d}{dx}v$

where u = first function and v = second function.

Example

$\begin{gathered} \frac{d}{dx}(x.sinx)=sinx.\frac{d}{dx}(x)+x\frac{d}{dx}(sinx) \\ \frac{d}{dx}(x.sinx)=sinx.(1)+x.(cosx) \\ \frac{d}{dx}(x.sinx)=sinx+xcosx \end{gathered}$

Quotient Rule:

If two functions are given in the quotient form or division form then the derivative of these functions using the quotient rule is given as:

$\frac{d}{dx}(\frac{u}{v})=\frac{v\frac{d}{dx}u-u\frac{d}{dx}v}{v^2}$

Example:

$\begin{gathered} \frac{d}{dx}(\frac{sinx}{x})=\frac{x\frac{d}{dx}sinx-sinx\frac{d}{dx}x}{x^2} \\ \frac{d}{dx}(\frac{sinx}{x})=\frac{x(cosx)-sinx(1)}{x^2} \\ \frac{d}{dx}(\frac{sinx}{x})=\frac{xcosx-sinx}{x^2} \end{gathered}$

Final Answer:

As explained in the explanation part.