Question
Find each derivative 2) \( y=\tan \left(x^{2}+3 x+2\right) \)
Ask by Sullivan Hampton. in the United States
Feb 26,2025
Upstudy AI Solution
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Answer
The derivative of \( y = \tan(x^{2} + 3x + 2) \) is \( y' = (2x + 3) \sec^{2}(x^{2} + 3x + 2) \).
Solution
We are given y = tan(f(x)) where f(x) = x² + 3x + 2. To find y′, apply the chain rule.
Step 1: Differentiate tan(f(x)):
d/dx [tan(f(x))] = sec²(f(x)) · f ′(x).
Step 2: Find f ′(x) where f(x) = x² + 3x + 2:
f ′(x) = 2x + 3.
Step 3: Substitute into the chain rule formula:
y′ = sec²(x² + 3x + 2) · (2x + 3).
Thus, the derivative is:
y′ = (2x + 3) sec²(x² + 3x + 2).
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Simplify this solution Beyond the Answer
To find the derivative of \( y=\tan \left(x^{2}+3 x+2\right) \), we will use the chain rule. The derivative of \(\tan(u)\) is \(\sec^2(u) \cdot \frac{du}{dx}\), where \( u = x^2 + 3x + 2 \). First, we differentiate \( u \): \[ \frac{du}{dx} = 2x + 3 \] Now we apply the chain rule: \[ \frac{dy}{dx} = \sec^2(u) \cdot \frac{du}{dx} = \sec^2(x^2 + 3x + 2) \cdot (2x + 3) \] Thus, the derivative is: \[ \frac{dy}{dx} = \sec^2(x^2 + 3x + 2) \cdot (2x + 3) \]
