Practice: Differentiation Of Logarithmic Equations

Let $k(x)=\ln (x),g(x)=\sqrt{x}$ and $h(x)=x^2-2$ , then $k(g(h(x)))=\ln \left( \sqrt{ x^2-2} \right)$ .

By Chain Rule ,

$\begin{aligned} \frac{d}{dx}k(g(h(x))) & = k'(g(h(x))) \cdot g'(h(x)) \cdot h'(x) \\ \frac{d}{dx} \ln \left( \sqrt{ x^2-2} \right) & = \frac{1}{\sqrt{x^2-2}}\cdot \frac{1}{2\sqrt{x^2-2}}\cdot 2x \\ & = \frac{x}{x^2-2} \\ f'(7) & =\frac{7}{47} \\ \Rightarrow 7+47 &=\boxed{54} \end{aligned}$