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