Rodriguez Guerrero
03/23/2024 · Middle School
If \( f(x)=\sin \left(x^{2}\right) \), find \( f^{\prime}(x) \). Find \( f^{\prime}(2) \).
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Tutor-Verified Answer
Step-by-step Solution
Find the first order derivative with respect to \( x \) for \( \sin(x^2) \).
Evaluate the derivative by following steps:
- step0: Evaluate the derivative:
\(\frac{d}{dx}\left(\sin\left(x^{2}\right)\right)\)
- step1: Use differentiation rules:
\(\frac{d}{dg}\left(\sin\left(g\right)\right)\times \frac{d}{dx}\left(x^{2}\right)\)
- step2: Find the derivative:
\(\cos\left(g\right)\times \frac{d}{dx}\left(x^{2}\right)\)
- step3: Calculate:
\(\cos\left(g\right)\times 2x\)
- step4: Substitute back:
\(\cos\left(x^{2}\right)\times 2x\)
- step5: Multiply the terms:
\(2x\cos\left(x^{2}\right)\)
Substitute \( x=2 \) into the expression \( 2x\cos(x^{2}) \).
Evaluate the expression by following steps:
- step0: Evaluate:
\(2x\cos\left(x^{2}\right)\)
- step1: Substitute:
\(2\times 2\cos\left(2^{2}\right)\)
- step2: Multiply the terms:
\(4\cos\left(2^{2}\right)\)
The derivative of \( f(x)=\sin(x^{2}) \) is \( f'(x) = 2x\cos(x^{2}) \).
Substitute \( x=2 \) into the derivative expression, we get \( f'(2) = 4\cos(2^{2}) \).
Quick Answer
\( f'(x) = 2x\cos(x^{2}) \), \( f'(2) = 4\cos(4) \)
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