Question

calculate the derivative of the following function.
y = (3x^5 - 4x^3 + 4)^25
dy/dx = □

Explanation:

Step1: Identify the outer - inner functions

Let $u = 3x^{5}-4x^{3}+4$, then $y = u^{25}$.

Step2: Differentiate the outer function

The derivative of $y$ with respect to $u$ is $\frac{dy}{du}=25u^{24}$ according to the power - rule $\frac{d}{du}(u^{n})=nu^{n - 1}$.

Step3: Differentiate the inner function

The derivative of $u$ with respect to $x$ is $\frac{du}{dx}=(3\times5x^{4}-4\times3x^{2})=15x^{4}-12x^{2}$ using the power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$.

Step4: Apply the chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $u = 3x^{5}-4x^{3}+4$, $\frac{dy}{du}=25u^{24}$, and $\frac{du}{dx}=15x^{4}-12x^{2}$ into the chain - rule formula.
$\frac{dy}{dx}=25(3x^{5}-4x^{3}+4)^{24}(15x^{4}-12x^{2})$

Answer:

$25(3x^{5}-4x^{3}+4)^{24}(15x^{4}-12x^{2})$