Question

calculate the derivative of the following function.
y = sin (7x^2 + 4x + 3)

Explanation:

Step1: Identify outer - inner functions

Let $u = 7x^{2}+4x + 3$, then $y=\sin(u)$.

Step2: Apply chain - rule

The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. First, find $\frac{dy}{du}$. Since $y = \sin(u)$, $\frac{dy}{du}=\cos(u)$. Second, find $\frac{du}{dx}$. Since $u = 7x^{2}+4x + 3$, $\frac{du}{dx}=14x + 4$.

Step3: Substitute back $u$

$\frac{dy}{dx}=\cos(7x^{2}+4x + 3)\cdot(14x + 4)$

Answer:

$\frac{dy}{dx}=\cos(7x^{2}+4x + 3)(14x + 4)$