Question

question given the function y = 3cos(6 + 6x^3), find dy/dx. answer attempt 1 out of 2 dy/dx =

Explanation:

Step1: Apply chain - rule

Let $u = 6+6x^{3}$, then $y = 3\cos(u)$. The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$.

Step2: Find $\frac{dy}{du}$

Differentiate $y = 3\cos(u)$ with respect to $u$. Using the derivative formula $\frac{d}{du}(\cos(u))=-\sin(u)$, we get $\frac{dy}{du}=- 3\sin(u)$.

Step3: Find $\frac{du}{dx}$

Differentiate $u = 6 + 6x^{3}$ with respect to $x$. Using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$, we have $\frac{du}{dx}=18x^{2}$.

Step4: Calculate $\frac{dy}{dx}$

Substitute $\frac{dy}{du}$ and $\frac{du}{dx}$ into the chain - rule formula: $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}=(-3\sin(u))\cdot(18x^{2})$. Replace $u$ with $6 + 6x^{3}$, so $\frac{dy}{dx}=-54x^{2}\sin(6 + 6x^{3})$.

Answer:

$-54x^{2}\sin(6 + 6x^{3})$