Question
Find the derivative using the appropriate rule or combination of rules. \( y=\tan \left(x^{2}+18 x\right) \) \( y^{\prime}=\square \)
Ask by Adkins Craig. in the United States
Mar 09,2025
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Answer
\( y' = (2x + 18) \sec^2(x^2 + 18x) \)
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The Deep Dive
To find the derivative of \( y = \tan(x^2 + 18x) \), we can use the chain rule. The derivative of \( \tan(u) \) is \( \sec^2(u) \cdot u' \), where \( u = x^2 + 18x \). First, we find the derivative of \( u \): \[ u' = \frac{d}{dx}(x^2 + 18x) = 2x + 18. \] Now applying the chain rule: \[ y' = \sec^2(u) \cdot u' = \sec^2(x^2 + 18x) \cdot (2x + 18). \] So, the final derivative is: \[ y' = \sec^2(x^2 + 18x)(2x + 18). \]