Question

this is section 2.8 problem 18: given y = u^2 and u = ln x;
$\frac{dy}{du}=$
$\frac{du}{dx}=$
$\frac{dy}{dx}=$
hint: follow examples 2 and 4.

Explanation:

Step1: Differentiate y with respect to u

Given $y = u^{2}$, using the power - rule $\frac{d}{du}(u^{n})=nu^{n - 1}$, we have $\frac{dy}{du}=2u$.

Step2: Differentiate u with respect to x

Given $u=\ln x$, using the derivative formula $\frac{d}{dx}(\ln x)=\frac{1}{x}$, we have $\frac{du}{dx}=\frac{1}{x}$.

Step3: Use the chain - rule

The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substituting $\frac{dy}{du}=2u$ and $\frac{du}{dx}=\frac{1}{x}$ into the chain - rule formula, and since $u = \ln x$, we get $\frac{dy}{dx}=2u\cdot\frac{1}{x}=2\ln x\cdot\frac{1}{x}=\frac{2\ln x}{x}$.

Answer:

$\frac{dy}{du}=2u$
$\frac{du}{dx}=\frac{1}{x}$
$\frac{dy}{dx}=\frac{2\ln x}{x}$