How do you find the derivative of $y = 3^{2 x + 7}$ $y= 3^(2x+7)$ ?

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How do you find the derivative of $y = 3^{2 x + 7}$ $y= 3^(2x+7)$ ?

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Answer 1 · 2016-04-29T00:02:05

$\frac{d y}{d x} = 3^{2 x + 7} ln (9)$ $dy/dx=3^(2x+7)ln(9)$

Explanation:

Using the derivative $\frac{d}{d x} a^{x} = a^{x} ln (a)$ $d/dxa^x = a^xln(a)$ together with the chain rule, we can note that $3^{2 x + 7}$ $3^(2x+7)$ = $f (g (x))$ $f(g(x))$ where $f (x) = 3^{x}$ $f(x) = 3^x$ and $g (x) = 2 x + 7$ $g(x)=2x+7$ and apply the chain rule to obtain:

$\frac{d y}{d x} = \frac{d}{d x} 3^{2 x + 7}$ $dy/dx = d/dx3^(2x+7)$

$= \frac{d}{d x} f (g (x))$ $=d/dxf(g(x))$

$=f'(g(x))g'(x)$

$=3^(2x+7)ln(3)*2$

$=3^(2x+7)ln(9)$

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How do you find the derivative of $y = 3^{2 x + 7}$ $y= 3^(2x+7)$ ?

1 Answer

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