Question

What is the derivative of `y=x^(5x)`?

Answer 1

`\frac{d}{dx}(x^{5x})=5x^{5x}(\ln (x)+1)`

Explanation:

`\frac{d}{dx}(x^{5x})`

Applying exponent rule,
`a^b=e^{b\ln (a)}`

`x^{5x}=e^{5x\ln (x)}`

`=\frac{d}{dx}(e^{5x\ln (x)})`

Applying chain rule,
`\frac{df(u)}{dx}=\frac{df}{du}\cdot \frac{du}{dx}`

Let `5x\ln (x)=u`

`=\frac{d}{du}(e^u)\frac{d}{dx}(5x\ln (x))`

We know,
`\frac{d}{du}(e^u)=e^u`
and,
`\frac{d}{dx}(5x\ln (x))=5(\ln (x)+1)`

So,
`\frac{d}{dx}(5x\ln (x))=5(\ln (x)+1)`

Substituting back,
`u=5x\ln (x)`

Simplifying,
`\frac{d}{dx}(x^{5x})=5x^{5x}(\ln (x)+1)`