Question

How do you differentiate `g(x) =e^x*1/x^2` using the product rule?

Answer 1

`\frac{e^x}{x^2}-\frac{2e^x}{x^3}`

Explanation:

`\frac{d}{dx}(e^x\frac{1}{x^2})`

Applying product rule,`(fcdot g)^'=f^'cdot g+fcdot g^'`

`f=e^x,g=\frac{1}{x^2}`

`=\frac{d}{dx}\(e^x\)\frac{1}{x^2}+\frac{d}{dx}\(\frac{1}{x^2}\)e^x`

We know,
`\frac{d}{dx}\(e^x\)=e^x` and

`\frac{d}{dx}\(\frac{1}{x^2}\)=-\frac{2}{x^3}`

So,
`=e^x\frac{1}{x^2}+\(-\frac{2}{x^3}\)e^x`

Finally, simplifying it,
`=\frac{e^x}{x^2}-\frac{2e^x}{x^3}`