Question

given y = f(u) and u = g(x), find $\frac{dy}{dx}=f(g(x))g(x)$ for the following functions. y = sin u, u = 6x + 5 $\frac{dy}{dx}=f(g(x))g(x)=square$

Explanation:

Step1: Find the derivative of $y = f(u)$ with respect to $u$

The derivative of $y=\sin u$ with respect to $u$ is $y^\prime=f^\prime(u)=\cos u$.

Step2: Find the derivative of $u = g(x)$ with respect to $x$

The derivative of $u = 6x + 5$ with respect to $x$ is $u^\prime=g^\prime(x)=6$.

Step3: Apply the chain - rule $\frac{dy}{dx}=f^\prime(g(x))g^\prime(x)$

Substitute $u = g(x)=6x + 5$ into $f^\prime(u)$ and multiply by $g^\prime(x)$. We get $f^\prime(g(x))=\cos(6x + 5)$ and $g^\prime(x)=6$. So $\frac{dy}{dx}=6\cos(6x + 5)$.

Answer:

$6\cos(6x + 5)$