hw 14 - derivatives of logs section 2.9: problem 4 (1 point) compute the derivatives of the given functions…

hw 14 - derivatives of logs section 2.9: problem 4 (1 point) compute the derivatives of the given functions. a) $f(x)=5ln x$. $f(x)=square$. b) $g(x)=ln(x^{2})$. $g(x)=square$

hw 14 - derivatives of logs section 2.9: problem 4 (1 point) compute the derivatives of the given functions. a) $f(x)=5ln x$. $f(x)=square$. b) $g(x)=ln(x^{2})$. $g(x)=square$

Answer

Explanation:

Step1: Recall derivative rule for $\ln x$

The derivative of $\ln x$ is $\frac{1}{x}$. For $y = 5\ln x$, using the constant - multiple rule of differentiation $(cf(x))'=cf'(x)$ where $c = 5$ and $f(x)=\ln x$. $f'(x)=5\times\frac{1}{x}=\frac{5}{x}$

Step2: Recall chain - rule and power - rule for $\ln(x^{2})$

Let $u = x^{2}$, then $g(x)=\ln(u)$. By the chain - rule $\frac{dg}{dx}=\frac{dg}{du}\cdot\frac{du}{dx}$. The derivative of $\ln(u)$ with respect to $u$ is $\frac{1}{u}$, and the derivative of $u = x^{2}$ with respect to $x$ is $2x$. Substituting $u = x^{2}$ back in, we get $g'(x)=\frac{1}{x^{2}}\cdot2x=\frac{2}{x}$.

Answer:

a. $\frac{5}{x}$ b. $\frac{2}{x}$