Question

Differentiate with respect to x:-
5^3 cos x-2​

Answer 1

$\textbf{Given:}$

$5^{3\,cosx}-2$

$\textbf{To find:}$

$\text{Derivative of 5^{3\,cosx}-2 }$

$\textbf{Solution:}$

$\text{We apply chain rule to differentiate the given function}$

$\boxed{\dfrac{d[f(g(x))]}{dx}=f'(g(x))\,g'(x)}$

$\text{Let,}\,y=5^{3\,cosx}-2$

$\implies\,y=(5^3)^{cosx}-2$

$\implies\,y=125^{cosx}-2$

$\text{Differentiate with resepect to 'x'}$

$\dfrac{dy}{dx}=\dfrac{d(125^{cosx})}{dx}-\dfrac{d(2)}{dx}$

$\text{Using,}$

$\boxed{\dfrac{d(a^x)}{dx}=a^x\,loga}$

$\boxed{\dfrac{d(c)}{dx}=0\;\;\text{where c is constant}}$

$\dfrac{dy}{dx}=125^{cosx}\,log\,125\,\dfrac{d(cosx)}{dx}-0$

$\dfrac{dy}{dx}=125^{cosx}\,log\,125(-sinx)$

$\dfrac{dy}{dx}=-125^{cosx}\,log\,125\,sinx$

$\textbf{Answer:}$

$\textbf{Derivative of \bf\,5^{3\,cosx}-2 is \bf\,-125^{cosx}\,log\,125\,sinx }$

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