Question

find $\frac{dy}{dx}$ for the following function.
y = 5 cos x sin x
$\frac{dy}{dx}=square$

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u\cdot v$, then $\frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}$. Here, $u = 5\cos x$ and $v=\sin x$.

Step2: Find derivatives of $u$ and $v$

The derivative of $u = 5\cos x$ with respect to $x$ is $\frac{du}{dx}=- 5\sin x$, and the derivative of $v=\sin x$ with respect to $x$ is $\frac{dv}{dx}=\cos x$.

Step3: Substitute into product - rule

$\frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}=(5\cos x)\cos x+(\sin x)(-5\sin x)$.

Step4: Simplify the expression

$\frac{dy}{dx}=5\cos^{2}x - 5\sin^{2}x=5(\cos^{2}x-\sin^{2}x)$. Using the double - angle formula $\cos(2x)=\cos^{2}x - \sin^{2}x$, we get $\frac{dy}{dx}=5\cos(2x)$.

Answer:

$5\cos(2x)$