Mathematics

Mathematics

Problem

For y=sin⁡(5x3)y=\sin(5x^{3}), answer the following: a. (i) Find dydx\dfrac{dy}{dx}. b. (ii) Hence find d2ydx2\dfrac{d^{2}y}{dx^{2}}.
Intuitive explanation

The key here is the chain rule — you're differentiating a composite function where the outer function is sine and the inner function is a polynomial. Start by treating 5x35x^3 as a single "thing," apply the derivative of sine (which gives cosine), then multiply by the derivative of that inner polynomial. This layering is what makes chain rule problems work: peel back one layer at a time.

Intuitive explanation

The key here is the chain rule — you're differentiating a composite function where the outer function is sine and the inner function is a polynomial. Start by treating 5x35x^3 as a single "thing," apply the derivative of sine (which gives cosine), then multiply by the derivative of that inner polynomial. This layering is what makes chain rule problems work: peel back one layer at a time.

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