Question

this is similar to section 2.8 problem 22: for the function y = 2ln(8x^5 + 5x^3 + 6), find $\frac{dy}{dx}$. answer: $\frac{dy}{dx}=$ . hint: follow example 5. resources ebook

Explanation:

Step1: Apply chain - rule

Let $u = 8x^{5}+5x^{3}+6$, then $y = 2\ln(u)$. The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. First, find $\frac{dy}{du}$. The derivative of $y = 2\ln(u)$ with respect to $u$ is $\frac{dy}{du}=\frac{2}{u}$.

Step2: Find $\frac{du}{dx}$

Differentiate $u = 8x^{5}+5x^{3}+6$ with respect to $x$. Using the power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, we have $\frac{du}{dx}=40x^{4}+15x^{2}$.

Step3: Calculate $\frac{dy}{dx}$

Substitute $\frac{dy}{du}$ and $\frac{du}{dx}$ into the chain - rule formula: $\frac{dy}{dx}=\frac{2}{u}\cdot(40x^{4}+15x^{2})$. Replace $u$ with $8x^{5}+5x^{3}+6$. So, $\frac{dy}{dx}=\frac{2(40x^{4}+15x^{2})}{8x^{5}+5x^{3}+6}=\frac{80x^{4}+30x^{2}}{8x^{5}+5x^{3}+6}$.

Answer:

$\frac{80x^{4}+30x^{2}}{8x^{5}+5x^{3}+6}$