Question
find \( \frac{d y}{d x} \) 2) \( y=\sec ^{2}\left(3 x^{2}\right. \) 3) \( y \)
Ask by Norton Frazier. in the United States
Mar 13,2025
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Answer
The derivative of \( y = \sec^{2}(3x^{2}) \) with respect to \( x \) is \( y' = 12x\sec^{2}(3x^{2})\tan(3x^{2}) \).
Solution
Find the first order derivative with respect to \( x \) for \( y=\sec^{2}(3x^{2}) \).
Evaluate the derivative by following steps:
- step0: Evaluate the derivative:
\(y^{\prime}=\frac{d}{dx}\left(\sec^{2}\left(3x^{2}\right)\right)\)
- step1: Use differentiation rules:
\(y^{\prime}=\frac{d}{dg}\left(g^{2}\right)\times \frac{d}{dx}\left(\sec\left(3x^{2}\right)\right)\)
- step2: Find the derivative:
\(y^{\prime}=2g\times \frac{d}{dx}\left(\sec\left(3x^{2}\right)\right)\)
- step3: Calculate:
\(y^{\prime}=2g\times 6\sec\left(3x^{2}\right)\tan\left(3x^{2}\right)\times x\)
- step4: Substitute back:
\(y^{\prime}=2\sec\left(3x^{2}\right)\times 6\sec\left(3x^{2}\right)\tan\left(3x^{2}\right)\times x\)
- step5: Multiply the terms:
\(y^{\prime}=12\sec\left(3x^{2}\right)\sec\left(3x^{2}\right)\tan\left(3x^{2}\right)\times x\)
- step6: Calculate:
\(y^{\prime}=12\sec^{2}\left(3x^{2}\right)\tan\left(3x^{2}\right)\times x\)
- step7: Reorder the terms:
\(y^{\prime}=12x\sec^{2}\left(3x^{2}\right)\tan\left(3x^{2}\right)\)
The first-order derivative of \( y = \sec^{2}(3x^{2}) \) with respect to \( x \) is \( y' = 12x\sec^{2}(3x^{2})\tan(3x^{2}) \).
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Simplify this solution Extra Insights
To find \( \frac{dy}{dx} \) for the function \( y = \sec^2(3x^2) \), we'll use the chain rule. First, recall that the derivative of \( \sec^2(u) \) with respect to \( u \) is \( 2\sec^2(u)\tan(u) \). Here \( u = 3x^2 \), so we need to first find \( \frac{du}{dx} = 6x \). Now we apply the chain rule: \[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 2\sec^2(3x^2)\tan(3x^2) \cdot 6x. \] Thus, \[ \frac{dy}{dx} = 12x \sec^2(3x^2) \tan(3x^2). \]
