Question

Find the derivative of sin x cos x​

Answer 1

Answer:

(x)=cosx(cosx)+sinx(−sinx)

f

′

(x)=cos

2

x−sin

2

x

Use the identity cos2x=cos

2

x−sin

2

x

f

′

(x)=cos2x

Step-by-step explanation:

Explanation: The product rule can be used to differentiate any function of the form f(x)=g(x)h(x) . It states that f'(x)=g'(x)h(x)+g(x)h'(x) . The derivative of sinx is cosx and the derivative of cosx is −sinx .

Answer 2

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$let \: y = f (x) = sinx \\ ⇉f(x + h) = \sin(x + h)$

$ʙʏ \: ᴅᴇғɪɴɪᴛɪᴏɴ \\ ⇒ \frac{dy}{dx} = \binom{lim}{x→0} \frac{f(x + h) - f(x)}{h} \\ \\ = \frac{ \sin(x + h) - sinx }{h} \\ = \frac{2 \cos( \frac{x + h + x}{2} ) . \sin( \frac{x + h - x}{2} ) }{2 \times \frac{h}{2} } \\ = \frac{ \cos( \frac{2x + h}{2} ) . \sin( \frac{h}{2} ) }{ \frac{h}{2} } \\ = \binom{lim}{x→0} \cos( \frac{2x + h}{2} ).1 \\ \frac{dy}{dx} sinx = cosx$

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