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## Solution to Mathematics for Machine Learning Exercise 5.1

Compute the derivative $f’(x)$ for $$f(x)=\log(x^4)\sin(x^3).$$ Solution: By Chain Rule (5.32), we have$$\big(\sin(x^3)\big)’=\cos(x^3)(x^3)’=3x^2\cos(x^3).$$We also have$$\big(\log(x^4)\big)’=\big(4\log(x)\big)’=\frac{4}{x}=4x^{-1}.$$Applying Product Rule (5.29), we obtain\begin{align*} f’(x)=&\ \log(x^4) \sin(x^3)\\=&\ \big(\log(x^4)\big)’\sin(x^3)+\log(x^4)\big(\sin(x^3)\big)’\\=&\ 4x^{-1}\sin(x^3)+\log(x^4)3x^2\cos(x^3)\\=&\…