differentiate the function using one or more of the differentiation rules. y = (8 + 3x^5)^6 y = □

Explanation:

Step1: Identify the outer - inner functions

Let $u = 8 + 3x^{5}$, so $y = u^{6}$.

Step2: Differentiate the outer function

The derivative of $y$ with respect to $u$ is $\frac{dy}{du}=6u^{5}$ using the power rule $\frac{d}{du}(u^{n})=nu^{n - 1}$ with $n = 6$.

Step3: Differentiate the inner function

The derivative of $u$ with respect to $x$ is $\frac{du}{dx}=15x^{4}$ using the power rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$ for $a = 3$ and $n = 5$ and $\frac{d}{dx}(c)=0$ for $c = 8$.

Step4: Apply the chain rule

By the chain rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}=6u^{5}$ and $\frac{du}{dx}=15x^{4}$ into the chain - rule formula. Replace $u$ with $8 + 3x^{5}$. So $\frac{dy}{dx}=6(8 + 3x^{5})^{5}\cdot15x^{4}=90x^{4}(8 + 3x^{5})^{5}$.

Answer:

$90x^{4}(8 + 3x^{5})^{5}$