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  4. If y=2a x and (d y/d x)= log 256 at x=1, then find value of a

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Q. If $y=2^{a x}$ and $\frac{d y}{d x}=\log 256$ at $x=1$, then find value of $a$

Continuity and Differentiability

Solution:

$\log y=a x \log 2$
$\frac{1}{y} \frac{d y}{d x}=a \log 2$
$\frac{d y}{d x}=2^{a x} \cdot a \log 2$
$\left.\frac{d y}{d x}\right)_{x=1}=2^{a} \cdot a \log 2=8 \log 2$
$=2^{a} \cdot a=8 ; a=2$