Q.Find the first and second derivatives of $y = 2x^5 - 3x^3 + 4x$.

The first derivative is found by the power rule applied term by term; differentiating that result once more, again by the power rule, gives the second derivative.

Given: $y=2x^5-3x^3+4x$.

Step 1 — Differentiate once (power rule term by term): $\dfrac{d}{dx}(2x^5)=10x^4$; $\dfrac{d}{dx}(-3x^3)=-9x^2$; $\dfrac{d}{dx}(4x)=4$. So $\dfrac{dy}{dx}=10x^4-9x^2+4$.

Step 2 — Differentiate the result again: $\dfrac{d}{dx}(10x^4)=40x^3$; $\dfrac{d}{dx}(-9x^2)=-18x$; $\dfrac{d}{dx}(4)=0$ (a constant). So $\dfrac{d^2y}{dx^2}=40x^3-18x$. …

Stopping after the first derivative when a question specifically asks for the second derivative, or differentiating the ORIGINAL function twice independe …