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\)`