Question

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

Explanation:

Step1: Identify outer - inner functions

Let $u = 7x^{2}+4x + 3$, 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 $u = 7x^{2}+4x + 3$ with respect to $x$ is $\frac{du}{dx}=14x + 4$.

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 $\frac{du}{dx}=14x + 4$ back in, and replace $u$ with $7x^{2}+4x + 3$. So $\frac{dy}{dx}=\cos(7x^{2}+4x + 3)\cdot(14x + 4)=(14x + 4)\cos(7x^{2}+4x + 3)$.

Answer:

$(14x + 4)\cos(7x^{2}+4x + 3)$