Q.Differentiate $y = (3x^2 + 5)^4$ with respect to $x$.

This is a power applied to a polynomial — a composite function — so use the chain rule.

With inner $u=3x^2+5$ ($u'=6x$): $\tfrac{dy}{dx}=4(3x^2+5)^3\cdot 6x=24x(3x^2+5)^3$.

The function is a composite: an outer fourth power applied to the inner polynomial $u = 3x^2 + 5$. Apply the chain rule (§3).

Identify inner and outer. Inner $u = 3x^2 + 5$, outer $y = u^4$.

Differentiate the outer with respect to $u$. $\dfrac{dy}{du} = 4u^3 = 4(3x^2+5)^3$.

Differentiate the inner with respect to $x$. $\dfrac{du}{dx} = 6x$.

Multiply (chain rule) $\dfrac{dy}{dx} = \dfrac{dy}{du}\cdot\dfrac{du}{dx}$:

Check (dual-solve): verify numerically at $x = 1$. The formula gives $24(1)(3+5)^3 = 24 \cdot 8^3 = 24 \cdot 512 = 12288$. Directly, $y(1) = 8^4 = 4096$; using $y(1.001) = (8.006003)^4 \approx 4108.31$ gives slope $\approx \dfrac{4108.31 - 4096}{0.001} \approx 12310$, agreeing with $12288$ up to the small finite-difference error — confirming the derivative.

The dominant mistake is stopping at $4(3x^2+5)^3$ and forgetting to multiply by the inner derivative $6x$. Without the $\times\,6x$ the answer would be the derivative of $u^4$ with respect to $u$, not with respect to $x$. Always finish the chain rule with the inner function's derivative.