Question

calculate the derivative of the following function. y = sin(4x^2 + 6x + 2)
\frac{dy}{dx}=square

Explanation:

Step1: Identify outer - inner functions

Let $u = 4x^{2}+6x + 2$, then $y=\sin(u)$.

Step2: Differentiate outer function

The derivative of $y = \sin(u)$ with respect to $u$ is $\frac{dy}{du}=\cos(u)$.

Step3: Differentiate inner function

The derivative of $u = 4x^{2}+6x + 2$ with respect to $x$ is $\frac{du}{dx}=8x + 6$.

Step4: Apply chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}=\cos(u)$ and $\frac{du}{dx}=8x + 6$ back in, and replace $u$ with $4x^{2}+6x + 2$. So $\frac{dy}{dx}=\cos(4x^{2}+6x + 2)\cdot(8x + 6)$.

Answer:

$(8x + 6)\cos(4x^{2}+6x + 2)$