Differentiate w.r.t. x the function in Exercise: (5x)^ 3 2x - Vidyadip

Question

Differentiate w.r.t. x the function in Exercise:
$(5\text{x})^{3\cos2\text{x}}$

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Answer

Let $\text{y}=(5\text{x})^{3\cos2\text{x}}$

Taking logaritthm on both the sides, we obtain

$\log\text{y}=3\cos2\text{x}\log5\text{x}$

Differentiating both sides with respect to x, we obtain

$\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=3\Big[\log5\text{x}.\frac{\text{d}}{\text{dx}}(\cos2\text{x})+\cos2\text{x}.\frac{\text{d}}{\text{dx}}(\log5\text{x})\Big]$

$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=3\text{x}\Big[\log5\text{x}(-\sin2\text{x}).\frac{\text{d}}{\text{dx}}(2\text{x})+\cos2\text{x}.\frac{1}{5\text{x}}.\frac{\text{d}}{\text{dx}}(5\text{x})\Big]$

$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=3\text{y}\Big[-2\sin2\text{x}\log5\text{x}+\frac{\cos2\text{x}}{\text{x}}\Big]$

$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=3\text{y}\Big[\frac{3\cos2\text{x}}{\text{x}}-6\sin2\text{x}\log5\text{x}\Big]$

$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=(5\text{x})^{\text{x}\cos2\text{x}}\Big[\frac{3\cos2\text{x}}{\text{x}}-6\sin2\text{x}\log5\text{x}\Big]$

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