Derivatives with the product and quotient rules

Text

Compute the derivatives of the following functions, using the product or quotient rule:

• (a) $\big[x\ln(x)\big]'$
• (b) $\left[\dfrac{\ln(x)}{x^2+x}\right]'$
• (c) $\big[\sin(x)\,e^x\big]'$
• (d) $\left[\dfrac{\sin(x)}{\sqrt{x}}\right]'$

Solution

Product rule $(fg)'=f'g+fg'$ and quotient rule $\left(\dfrac{f}{g}\right)'=\dfrac{f'g-fg'}{g^2}$.

• (a) $\big[x\ln x\big]' = 1\cdot\ln x + x\cdot\dfrac{1}{x} = \ln x + 1$
• (b) $\left[\dfrac{\ln x}{x^2+x}\right]' = \dfrac{\frac{1}{x}(x^2+x) - \ln x\,(2x+1)}{(x^2+x)^2} = \dfrac{(x+1) - (2x+1)\ln x}{(x^2+x)^2}$
• (c) $\big[\sin x\,e^x\big]' = \cos x\,e^x + \sin x\,e^x = e^x(\sin x + \cos x)$
• (d) $\left[\dfrac{\sin x}{\sqrt{x}}\right]' = \dfrac{\cos x\,\sqrt{x} - \sin x\cdot\frac{1}{2\sqrt{x}}}{x} = \dfrac{2x\cos x - \sin x}{2x\sqrt{x}}$ $\boxed{\ln x + 1;\ \ \dfrac{(x+1)-(2x+1)\ln x}{(x^2+x)^2};\ \ e^x(\sin x+\cos x);\ \ \dfrac{2x\cos x-\sin x}{2x\sqrt{x}}}$