Question

y = (cos 5x)^x
find y'

Answer 1

@campbell_st, the power rule doesn't work for variable powers.

Take the logarithm of \(y\) first:
\[\begin{align*}
y&=(\cos5x)^x\\\\
\ln y&=x\ln(\cos 5x)\\\\
\frac{dy}{dx}[\ln y(x)]&=\frac{d}{dx}[x]\ln(\cos5x)+x\frac{d}{dx}[\ln(\cos5x)]\\\\
\frac{1}{y}\frac{dy}{dx}&=\ln(\cos5x)+x\frac{\dfrac{d}{dx}[\cos5x]}{\cos5x}\\\\
\frac{1}{(\cos5x)^x}\frac{dy}{dx}&=\ln(\cos5x)-5x\frac{\sin5x}{\cos5x}\\\\
y'&=(\cos5x)^x\left(\ln(\cos5x)-5x\tan5x\right)
\end{align*}\]