differentiate the function using one or more of the differentiation rules. y=(2 + 5x^2)^9 y=□

Explanation:

Step1: Identify the outer - inner functions

Let $u = 2 + 5x^{2}$, then $y = u^{9}$.

Step2: Differentiate the outer function

The derivative of $y$ with respect to $u$ is $\frac{dy}{du}=9u^{8}$.

Step3: Differentiate the inner function

The derivative of $u$ with respect to $x$ is $\frac{du}{dx}=10x$.

Step4: Apply the chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $u = 2 + 5x^{2}$, $\frac{dy}{du}=9u^{8}$ and $\frac{du}{dx}=10x$ into the chain - rule formula. We get $\frac{dy}{dx}=9(2 + 5x^{2})^{8}\cdot10x$.

Answer:

$90x(2 + 5x^{2})^{8}$