Carroll Rose
07/12/2024 · Primary School
(a) Given \( f(x)=2 x \tan x \), where \( 0
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Step-by-step Solution
Find the first order derivative with respect to \( x \) for \( 2x\tan(x) \).
Evaluate the derivative by following steps:
- step0: Evaluate the derivative:
\(\frac{d}{dx}\left(2x\tan\left(x\right)\right)\)
- step1: Use differentiation rules:
\(\frac{d}{dx}\left(2\right)\times x\tan\left(x\right)+2\times \frac{d}{dx}\left(x\right)\tan\left(x\right)+2x\times \frac{d}{dx}\left(\tan\left(x\right)\right)\)
- step2: Find the derivative:
\(0\times x\tan\left(x\right)+2\times \frac{d}{dx}\left(x\right)\tan\left(x\right)+2x\times \frac{d}{dx}\left(\tan\left(x\right)\right)\)
- step3: Calculate:
\(0+2\times \frac{d}{dx}\left(x\right)\tan\left(x\right)+2x\times \frac{d}{dx}\left(\tan\left(x\right)\right)\)
- step4: Calculate:
\(0+2\times 1\times \tan\left(x\right)+2x\times \frac{d}{dx}\left(\tan\left(x\right)\right)\)
- step5: Calculate:
\(0+2\tan\left(x\right)+2x\times \frac{d}{dx}\left(\tan\left(x\right)\right)\)
- step6: Calculate:
\(0+2\tan\left(x\right)+2x\sec^{2}\left(x\right)\)
- step7: Remove 0:
\(2\tan\left(x\right)+2x\sec^{2}\left(x\right)\)
Substitute \( x=\frac{\pi}{4} \) into the expression \( 2\tan(x)+2x\sec^{2}(x) \).
Evaluate the expression by following steps:
- step0: Evaluate:
\(2\tan\left(x\right)+2x\sec^{2}\left(x\right)\)
- step1: Substitute:
\(2\tan\left(\frac{\pi }{4}\right)+2\times \frac{\pi }{4}\sec^{2}\left(\frac{\pi }{4}\right)\)
- step2: Calculate:
\(2\times 1+2\times \frac{\pi }{4}\sec^{2}\left(\frac{\pi }{4}\right)\)
- step3: Multiply:
\(2+2\times \frac{\pi }{4}\sec^{2}\left(\frac{\pi }{4}\right)\)
- step4: Multiply the terms:
\(2+\frac{\pi }{2}\sec^{2}\left(\frac{\pi }{4}\right)\)
- step5: Transform the expression:
\(2+\frac{\pi }{2}\cos^{-2}\left(\frac{\pi }{4}\right)\)
- step6: Transform the expression:
\(2+\pi \)
The derivative of the function \( f(x)=2x\tan(x) \) is \( f'(x) = 2\tan(x) + 2x\sec^{2}(x) \).
Substitute \( x=\frac{\pi}{4} \) into the derivative expression, we get \( f'\left(\frac{\pi}{4}\right) = 2 + \pi \).
Quick Answer
\( f'\left(\frac{\pi}{4}\right) = 2 + \pi \)
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