Question

Calculate the derivative of the following functions. $$y=\sin ^{2}\left(e^{3 x+1}\right)$$

Short answer

Question: Find the derivative of the function $$y = \sin^2(e^{3x+1})$$. Answer: The derivative of the function $$y = \sin^2(e^{3x+1})$$ is $$\frac{dy}{dx} = 6e^{3x+1}\sin(e^{3x+1})\cos(e^{3x+1})$$.

Step-by-step solution

Identify the outer and inner functions

We can identify $$f(u) = \sin^2(u)$$ as the outer function and inner function $$u = e^{3x+1}$$. Our aim is to find $$\frac{dy}{dx} = \frac{d(\sin^2(u))}{dx} = \frac{d(\sin^2(u))}{du} \cdot \frac{du}{dx}$$.

Differentiate the outer function

To differentiate the outer function, we have to apply the power rule to $$y=f(u)=\sin^2(u)$$, and then, we will apply the chain rule to the resulting expression: $$\frac{d(\sin^2(u))}{du} = 2\sin(u)\cos(u)$$

Differentiate the inner function

Now, we will differentiate the inner function $$u = e^{3x+1}$$ with respect to x: $$\frac{du}{dx} = \frac{d(e^{3x+1})}{dx} = 3e^{3x+1}$$

Apply the chain rule

Finally, we will apply the chain rule to find the derivative of y with respect to x: $$\frac{dy}{dx} = \frac{d(\sin^2(u))}{du} \cdot \frac{du}{dx} = (2\sin(u)\cos(u)) \cdot (3e^{3x+1})$$

Replace u with the inner function

Now we will replace u with the inner function $$u = e^{3x+1}$$: $$\frac{dy}{dx} = 2\sin(e^{3x+1})\cos(e^{3x+1}) \cdot 3e^{3x+1}$$

Simplify the expression

Finally, we simplify the expression to get the derivative of the given function: $$\frac{dy}{dx} = 6e^{3x+1}\sin(e^{3x+1})\cos(e^{3x+1})$$