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Derivatives & Second-Order Derivatives · Question 22 of 96

Question 22

MediumOriginal question

What is ddx(sin⁡xcos⁡x)\frac{d}{dx}\left(\sin x\cos x\right)?

Options for question 22

Show the answer and explanation
CorrectNot quiteAnswer A

Correct answer: A

Explanation

  1. Product rule: cos⁡x⋅cos⁡x+sin⁡x⋅(−sin⁡x)=cos⁡2x−sin⁡2x\cos x \cdot \cos x + \sin x \cdot (-\sin x) = \cos^{2} x - \sin^{2} x.

  2. cos⁡2x−sin⁡2x=cos⁡2x\cos^{2} x - \sin^{2} x = \cos 2x. (Or first write sin⁡xcos⁡x=12sin⁡2x\sin x\cos x = \frac{1}{2}\sin 2x.)

Worth noting

−sin⁡xcos⁡x-\sin x\cos x multiplies the two derivatives, cos⁡x\cos x and −sin⁡x-\sin x, which is not the product rule.

Concept
Simplifying before or after differentiating
Typical time
90 s
Your time
—
The rule this testsThe rules of differentiation
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