What is the derivative of ${({(8^{x})}^{5})}^{x}$ $((8^x)^5)^x$ ?

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Answer 1 · 2016-06-10T19:50:00

$10 x {({(8^{x})}^{5})}^{x} ln 8$ $10x((8^x)^5)^x ln8$

${({(8^{x})}^{5})}^{x} = 8^{5 x^{2}}$ $((8^x)^5)^x = 8^(5x^2)$ .

So,

$\frac{d}{d x} ({({(8^{x})}^{5})}^{x}) = \frac{d}{d x} (8^{5 x^{2}})$ $d/dx(((8^x)^5)^x ) = d/dx(8^(5x^2))$ .

Now use $\frac{d}{d x} (8^{u}) = 8^{u} ln u \frac{d}{d x} (u)$ $d/dx(8^u) = 8^u lnu d/dx(u)$ to get

$\frac{d}{d x} (8^{5 x^{2}}) = 8^{5 x^{2}} ln 8 \frac{d}{d x} (5 x^{2})$ $d/dx(8^(5x^2)) = 8^(5x^2) ln8 d/dx(5x^2)$

$= 8^{5 x^{2}} ln 8 (10 x)$ $= 8^(5x^2) ln8 (10x)$

$= 10 x 8^{5 x^{2}} ln 8$ $= 10x 8^(5x^2) ln8$

What is the derivative of ((8^x)^5)^x((8x)5)x((8^x)^5)^x?