if (y=(5x)^{sin(4x)}), then (\frac{dy}{dx}=)

Explanation:

Step1: Take natural - log of both sides

$\ln y=\sin(4x)\ln(5x)$

Step2: Differentiate both sides with respect to (x)

Using the chain - rule and product - rule. The derivative of the left - hand side is (\frac{1}{y}\frac{dy}{dx}). For the right - hand side, by the product rule ((uv)^\prime = u^\prime v+uv^\prime) where (u = \sin(4x)) and (v=\ln(5x)). (u^\prime=\cos(4x)\cdot4) and (v^\prime=\frac{1}{x}). So (\frac{1}{y}\frac{dy}{dx}=4\cos(4x)\ln(5x)+\frac{\sin(4x)}{x})

Step3: Solve for (\frac{dy}{dx})

Multiply both sides by (y=(5x)^{\sin(4x)}) (\frac{dy}{dx}=(5x)^{\sin(4x)}\left(4\cos(4x)\ln(5x)+\frac{\sin(4x)}{x}\right))

Answer:

((5x)^{\sin(4x)}\left(4\cos(4x)\ln(5x)+\frac{\sin(4x)}{x}\right))