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Question: Differentiate each of the following w.r.t. x: \(\sqrt{\cos 3x}\)...

Differentiate each of the following w.r.t. x:
cos⁡3x\sqrt{\cos 3x}

Explanation

Solution

We solve this question by using the first principle of derivation. We differentiate the given function cos⁡3x\sqrt{\cos 3x} by differentiating the outer root function and then the inner function which is cos⁡3x.\cos 3x. This method of differentiation is used for composite functions as given above. Then we use the standard differentiation formula to solve this question.

Complete step by step answer:
In order to solve this question, let us first write down the given term needed to be differentiated with respect to x as,
⇒ddxcos⁡3x\Rightarrow \dfrac{d}{dx}\sqrt{\cos 3x}
Next, we know that the square root function is nothing but the value inside raised to the power 12.\dfrac{1}{2}. Thus, the above line can also be represented as,
⇒ddxcos⁡123x\Rightarrow \dfrac{d}{dx}{{\cos }^{\dfrac{1}{2}}}3x
We differentiate this using the method using the composite function differentiation or also known as chain rule. This is represented as f(x)=h(g(x)),ddxf(x)=ddxh(g(x)).ddxg(x).f\left( x \right)=h\left( g\left( x \right) \right),\dfrac{d}{dx}f\left( x \right)=\dfrac{d}{dx}h\left( g\left( x \right) \right).\dfrac{d}{dx}g\left( x \right). This is given by the differentiation of the outer function multiplied by the differentiation of the inner function. Using this for the above equation,
⇒12.cos⁡12−13xddx(cos⁡3x)\Rightarrow \dfrac{1}{2}.{{\cos }^{\dfrac{1}{2}-1}}3x\dfrac{d}{dx}\left( \cos 3x \right)
Simplifying the power of cos⁡3x,\cos 3x,
⇒12.cos⁡−123xddx(cos⁡3x)\Rightarrow \dfrac{1}{2}.{{\cos }^{-\dfrac{1}{2}}}3x\dfrac{d}{dx}\left( \cos 3x \right)
This negative half power indicated a square root term in the denominator. Also, we need use the chain rule again for cos⁡3x.\cos 3x. We know the differentiation of cos⁡t\cos t is given by, ddxcos⁡t=−sin⁡t.\dfrac{d}{dx}\cos t=-\sin t. Using this,
⇒12cos⁡3x.−sin⁡3xddx(3x)\Rightarrow \dfrac{1}{2\sqrt{\cos 3x}}.-\sin 3x\dfrac{d}{dx}\left( 3x \right)
We now differentiate 3x with respect to x to get 3.
⇒−sin⁡3x2cos⁡3x.3\Rightarrow \dfrac{-\sin 3x}{2\sqrt{\cos 3x}}.3
Hence, the derivative of cos⁡3x\sqrt{\cos 3x} with respect to x is −3sin⁡3x2cos⁡3x.\dfrac{-3\sin 3x}{2\sqrt{\cos 3x}}.

Note: We need to know the concept of chain rule in differentiation in order to solve the above sum. We also need to know basic formulae for differentiation to solve differentiation sums easily. Care must be taken to ensure the chain rule is applied else it will lead to a wrong answer.