Question

evaluate and simplify y.
y = tan(3x^3 + 2x + 7)
y = □

Explanation:

Step1: Identify the outer - inner functions

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

Step2: Differentiate the outer function

The derivative of $y = \tan(u)$ with respect to $u$ is $y_{u}'=\sec^{2}(u)$.

Step3: Differentiate the inner function

The derivative of $u = 3x^{3}+2x + 7$ with respect to $x$ is $u_{x}'=9x^{2}+2$.

Step4: Apply the chain - rule

By the chain - rule $y_{x}'=y_{u}'\cdot u_{x}'$. Substitute $y_{u}'$ and $u_{x}'$: $y'=\sec^{2}(3x^{3}+2x + 7)\cdot(9x^{2}+2)$.

Answer:

$(9x^{2}+2)\sec^{2}(3x^{3}+2x + 7)$