Question

differentiate the function using one or more of the differentiation rules.
y = (1 + 4x³)¹⁰
y = □

Explanation:

Step1: Apply the chain rule

Let \( u = 1 + 4x^{3}\), then \( y=u^{10}\).
The chain rule states that \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\).
First, find \(\frac{dy}{du}\):
If \( y = u^{10}\), then \(\frac{dy}{du}=10u^{9}\).

Step2: Find \(\frac{du}{dx}\)

If \( u = 1+4x^{3}\), then \(\frac{du}{dx}=12x^{2}\).

Step3: Substitute \( u\) back and multiply

Substitute \( u = 1 + 4x^{3}\) into \(\frac{dy}{du}\) and then multiply by \(\frac{du}{dx}\).
\(\frac{dy}{dx}=10(1 + 4x^{3})^{9}\cdot12x^{2}\).
Simplify the expression: \(\frac{dy}{dx}=120x^{2}(1 + 4x^{3})^{9}\).

Answer:

\(120x^{2}(1 + 4x^{3})^{9}\)