find the derivative of $y = 4x^{3}+3x + 10$

Explanation:

Step1: Differentiate each term

Use the power rule (\frac{d}{dx}(x^n)=nx^{n - 1}). For (y = 4x^{3}+3x + 10), (\frac{d}{dx}(4x^{3})=4\times3x^{3-1}=12x^{2}), (\frac{d}{dx}(3x)=3\times1x^{1 - 1}=3), (\frac{d}{dx}(10)=0) (since the derivative of a constant is (0)).

Step2: Combine the derivatives

By the sum rule (\frac{d}{dx}(u + v+w)=\frac{du}{dx}+\frac{dv}{dx}+\frac{dw}{dx}), so (\frac{dy}{dx}=\frac{d}{dx}(4x^{3})+\frac{d}{dx}(3x)+\frac{d}{dx}(10)=12x^{2}+3).

Answer:

(\frac{dy}{dx}=12x^{2}+3) (the first option)