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

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Answer 1 · 2016-05-30T10:53:43

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

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

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

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