# Exercises - The Product Rule

1. In each case, find $y'$

1. $\displaystyle{y = (x^2 - \frac{1}{x})(3x-4)}$

2. $\displaystyle{y = (\sqrt{x} + 5)(x^2 - 5)}$

3. $\displaystyle{y = (\sin x)(\cos x) - x \cos x + x^3}$

4. $\displaystyle{y = x^3 \sin x - x^3 \cos x}$

5. $\displaystyle{y = \ln e^x - e^x \ln x}$

2. Differentiate:

1. $\displaystyle{f\,(x) = (x^2+x)(3x+1)}$

2. $\displaystyle{f\,(x) = x^2e^x + \frac{1}{x^2}}$

3. $\displaystyle{f\,(x) = x^3\sin x + \sin (\frac{\pi}{4} )}$