Question

y = tan(5 * x + 4)
find $\frac{dy}{dx}$
$\frac{dy}{dx}=$
type sin(x) for sin($x$), cos(x) for cos($x$), and so on.
use x^2 to square x, x^3 to cube x, and so on.
use ( sin(x) )^2 to square sin(x).
do not simplify your answer.

Explanation:

Step1: Apply chain - rule

Let $u = 5x + 4$, then $y=\tan(u)$. The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$.

Step2: Differentiate $y$ with respect to $u$

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

Step3: Differentiate $u$ with respect to $x$

Since $u = 5x+4$, then $\frac{du}{dx}=5$.

Step4: Substitute $u$ and the derivatives back into the chain - rule formula

$\frac{dy}{dx}=\sec^{2}(5x + 4)\cdot5$.

Answer:

$5\sec^{2}(5x + 4)$