section 2.6: chain rule (homework) score: 140/170 answered: 14/17 progress saved done question 15 0/10 pts 4…

Explanation:

Step1: Rewrite the function

First, rewrite $y = f(x)=\ln\left(\sqrt[6]{x^{6}+3x^{4}+2}\right)$ as $y=\frac{1}{6}\ln\left(x^{6}+3x^{4}+2\right)$ using the property $\ln(a^{b})=b\ln(a)$.

Step2: Apply the chain - rule

The chain - rule states that if $y = \frac{1}{6}\ln(u)$ where $u=x^{6}+3x^{4}+2$, then $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. The derivative of $\frac{1}{6}\ln(u)$ with respect to $u$ is $\frac{1}{6u}$, and the derivative of $u = x^{6}+3x^{4}+2$ with respect to $x$ is $6x^{5}+12x^{3}$.

Step3: Calculate the derivative

Substitute $u$ back into the formula: $\frac{dy}{dx}=\frac{1}{6(x^{6}+3x^{4}+2)}\cdot(6x^{5}+12x^{3})$. Then simplify the expression: $\frac{6x^{5}+12x^{3}}{6(x^{6}+3x^{4}+2)}=\frac{x^{5}+2x^{3}}{x^{6}+3x^{4}+2}$.

Answer:

$\frac{x^{5}+2x^{3}}{x^{6}+3x^{4}+2}$