Question

Differentiate w.r.t. \(x\) the function: \((5x)^{3cos2x}\)

Answer 1

The correct answer is \((5x)^{3cos2x}[\frac{3cos\,2x}{x}-6sin\,2x\,log\,5x]\)
Let \(y=(5x)^{3cos2x}\)
Taking logarithm on both the sides, we obtain
\(logy=3cos\,2x\,log\,5x\)
Differentiating both sides with respect to \(x\), we obtain
\(\frac{1}{y}\frac{dy}{dx}=3[log5x.\frac{d}{dx}(cos2x)+cos2x\frac{d}{dx}(log5x)]\)
\(⇒\frac{dy}{dx}=3y[log5x(-sin2x).\frac{d}{dx}(2x)+cos2x.\frac{1}{5x}.\frac{d}{dx}(5x)]\)
\(⇒\frac{dy}{dx}=3y[-2sin2xlog5x+\frac{cos2x}{x}]\)
\(⇒\frac{dy}{dx}=3y[\frac{3cos\,2x}{x}-6sin\,2x\,log\,5x]\)
\(∴\frac{dy}{dx}=(5x)^{3cos2x}[\frac{3cos\,2x}{x}-6sin\,2x\,log\,5x]\)