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The derivative of sin2 x with respect to cos x is1. -2cos x2. -2sin x3. 2sin x4. None of these

Answer» Correct Answer - Option 1 : -2cos x

Concept:

Let u = f(x)  and v = g(x)

The derivative of u with respect to v is \(\rm \dfrac {du}{dv}\)

By chain Rule 

\(\rm \frac {du}{dv} = \dfrac {\frac {du}{dx}}{\frac {dv}{dx}}\)

 

Calculations:

Let u = sin2 x  and v = cos x

The derivative of sin2 x with respect to cos x is \(\rm \dfrac {du}{dv}\)

By chain Rule 

\(\rm \frac {du}{dv} = \dfrac {\frac {du}{dx}}{\frac {dv}{dx}}\)         ....(1)

On differentiating w.r.t x respectively, we get

⇒\(\rm \dfrac {du}{dx} = 2\sin x\; \cos x\) and \(\rm \dfrac {dv}{dx} = -\sin x\)

Equation (1) becomes,

\(\rm \dfrac {du}{dv} = \dfrac {2\sin x\cos x}{-\sin x}\)

⇒\(\rm \dfrac {du}{dv} = -2\cos x\)

Hence, the derivative of sin2 x with respect to cos x is - 2 cos x.


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