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Question: How do you differentiate \(\sin \left( \dfrac{x}{2} \right)\)....

How do you differentiate sin⁡(x2)\sin \left( \dfrac{x}{2} \right).

Explanation

Solution

Now to differentiate the function we will use chain rule of differentiation. According to chain rule of differentiation we have, Differentiation of f(g(x))f\left( g\left( x \right) \right) is f′(g(x)).g′(x)f'\left( g\left( x \right) \right).g'\left( x \right) . Now we know that the differentiation of sin⁡x\sin x is cos⁡x\cos x and d(cx)dx=c\dfrac{d\left( cx \right)}{dx}=c . Hence using this we will differentiate the given function.

Complete step-by-step solution:
Now let us understand the concept of differentiation.
Differentiation of a function gives us instantaneous rate of change of function with respect to the independent variable.
Differentiation of a function f with respect to x is shown as df(x)dx\dfrac{df\left( x \right)}{dx} or also sometimes as f′(x)f'\left( x \right)
Now consider the given function sin⁡x2\sin \dfrac{x}{2},
Now we know that the differentiation of a composite function of the type f(g(x))=f′(g(x)).g′(x)f\left( g\left( x \right) \right)=f'\left( g\left( x \right) \right).g'\left( x \right) .
Now here f(x)=sin⁡xf\left( x \right)=\sin x and g(x)=x2g\left( x \right)=\dfrac{x}{2}
Now we know that the differentiation of sin⁡x\sin x is cos⁡x\cos x and d(cx)dx=c\dfrac{d\left( cx \right)}{dx}=c hence differentiating the function we get,
f′(x)=cos⁡xf'\left( x \right)=\cos x and g′(x)=12g'\left( x \right)=\dfrac{1}{2}
Hence we get, f′(g(x))=cos⁡x2f'\left( g\left( x \right) \right)=\cos \dfrac{x}{2}
Now substituting the values in formula we have, the required differentiation as cos⁡x22\dfrac{\cos \dfrac{x}{2}}{2}
Hence the differentiation of the given function is cos⁡x22\dfrac{\cos \dfrac{x}{2}}{2}.

Note: Now note we have differentiation is nothing but instantaneous rate of change with respect to independent variables. Geometrically this is nothing but the slope of the curve. Now here we have differentiation of the given expression is cos⁡x22\dfrac{\cos \dfrac{x}{2}}{2} . Hence slope of the curve at any point x is given by cos⁡x22\dfrac{\cos \dfrac{x}{2}}{2} . Hence we can use differentiation to find the slope of the curve too.
Also note that here we have used chain rule of differentiation which is different from product rule. Not to be confused between f(g(x))f\left( g\left( x \right) \right) and f(x).g(x)f\left( x \right).g\left( x \right) . Differentiation of the function f.g is given by f′g+g′ff'g+g'f