Question

How do you find the derivative of `x^(7x)`?

Answer 1

`d/dx x^(7x)=7x^(7x)(ln(x)+1)`

Explanation:

Using the chain rule and the product rule, together with the following derivatives:

• `d/dx e^x = e^x`
• `d/dx ln(x) = 1/x`
• `d/dx x = 1`

we have

`d/dx x^(7x) = d/dx e^(ln(x^(7x)))`

`=d/dx e^(7xln(x))`

`=e^(7xln(x))(d/dx7xln(x))`

(by the chain rule with the functions `e^x` and `7xln(x)`)

`=7e^(ln(x^(7x)))(xd/dxln(x) + ln(x)d/dxx)`

(by the product rule, and factoring out the `7`)

`=7x^(7x)(x*1/x + ln(x)*1)`

`=7x^(7x)(ln(x)+1)`