Question #7e670

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Question #7e670

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Answer 1 · 2017-04-08T00:48:48

$\frac{21}{2} x^{\frac{21}{2} x} (ln (x) + 1)$ $21/2x^(21/2x)(ln(x)+1)$

Explanation:

$y = {\sqrt{x}}^{21 x}$ $y=sqrt(x)^(21x)$
$y = x^{\frac{21}{2} x}$ $y=x^(21/2x)$
$ln (y) = ln (x^{\frac{21}{2} x}) = \frac{21}{2} x ln (x)$ $ln(y)=ln(x^(21/2x))=21/2xln(x)$
Take the derivative with respect to $x$ $x$ for both sides:
$\frac{d}{d x} ln (y) = \frac{d}{d x} (\frac{21}{2} x ln (x))$ $d/dxln(y)=d/dx(21/2xln(x))$
Using the chain rule for the left-hand side and the product rule for the right-hand side:
$\frac{\frac{d y}{d x}}{y} = \frac{21}{2} (ln (x) + 1)$ $(dy/dx)/y=21/2(ln(x)+1)$
$\frac{d y}{d x} = \frac{21}{2} (ln (x) + 1) y$ $dy/dx=21/2(ln(x)+1)y$
Since $y = x^{\frac{21}{2} x}$ $y=x^(21/2x)$ :
$\frac{d y}{d x} = \frac{21}{2} x^{\frac{21}{2} x} (ln (x) + 1)$ $dy/dx=21/2x^(21/2x)(ln(x)+1)$

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Question #7e670

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