[-/1 Points] DETAILS
Find the derivative of the function.

Ask by Powers Todd. in the United States

Mar 20,2025

Question

[-/1 Points] DETAILS Find the derivative of the function. $$ y=\sin (\tan (7 x)) $$ $$ y^{\prime}=\square $$

UpStudy AI · Accepted Answer

Given 
$$
y = \sin\left(\tan(7x)\right),
$$
we differentiate by applying the chain rule multiple times.

1. Define the inner functions:
   - Let $$ u = \tan(7x) $$, so that $$ y = \sin(u) $$.
   - Further, let $$ v = 7x $$ so that $$ u = \tan(v) $$.

2. Differentiate $$ y = \sin(u) $$ with respect to $$ u $$:
   $$
   \frac{dy}{du} = \cos(u).
   $$

3. Differentiate $$ u = \tan(v) $$ with respect to $$ v $$:
   $$
   \frac{du}{dv} = \sec^2(v).
   $$

4. Differentiate $$ v = 7x $$ with respect to $$ x $$:
   $$
   \frac{dv}{dx} = 7.
   $$

5. Apply the chain rule:
   $$
   y' = \frac{dy}{du} \cdot \frac{du}{dv} \cdot \frac{dv}{dx} = \cos(u) \cdot \sec^2(v) \cdot 7.
   $$

6. Substitute back the expressions for $$ u $$ and $$ v $$:
   $$
   y' = 7 \cos\left(\tan(7x)\right) \sec^2(7x).
   $$

Thus, the derivative is:
$$
y' = 7 \cos\left(\tan(7x)\right) \sec^2(7x).
$$
The derivative of $$ y = \sin(\tan(7x)) $$ is $$ y' = 7 \cos(\tan(7x)) \sec^2(7x) $$.

[-/1 Points] DETAILS
Find the derivative of the function.

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