Question

Derivative of y=(5x^x)

Answer 1

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$given \: that \\ y = 5 {x}^{x} \\ let \: u = {x}^{x} \: \: \: \: \: \: ...(1) \\ taking \: log \: on \: both \: sides \\ we \: get \\ logu = log {x}^{x} \\ logu = xlogx \\ differentiating \: wrtx \\ \frac{1}{u} \frac{du}{dx} = logx + 1 \\ \frac{du}{dx} = u(1 + logx) \\ \frac{du}{dx} = {x}^{x} (1 + logx) \: \: \: \: \:... from(1) \\ \\ therefore \\ y = 5 {x}^{x} \\ y = 5u \\ differentiating \: wrtx \\ \frac{dy}{dx} = 5 \frac{du}{dx} \\ \frac{dy}{dx} = 5 {x}^{x} (1 + logx)$

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