Question

Differentiate $\sin \left( {\sin \,{x^2}} \right)$wrt $x$.

Answer 1

In this question we have to do the differentiation of a function $f(x)$which is in a form of composition function. So, for this we use the chain rule of the differentiation method. That rule states that if a function is Let given in a form of $fog$,then first differentiate function $'f'$and after that differentiate inner function and then multiply it. Now let us see in a mathematical form
$\dfrac{d}{{dx}}(fog) = \dfrac{{df}}{{dx}}.\dfrac{{dg}}{{dx}}$
For example, Differentiate ${e^{{x^2}}}$
$\Rightarrow$we want to find out the value of $\dfrac{{d{e^{{x^2}}}}}{{dx}}$
According to chain rule
$\dfrac{d}{{dx}}(fog) = \dfrac{{df}}{{dx}}.\dfrac{{dg}}{{dx}}$
$\Rightarrow \dfrac{{d{e^{x2}}}}{{dx}} = \dfrac{{d{e^{{x^2}}}}}{{dx}}.\dfrac{{d{x^2}}}{{dx}}$
$= {e^{{x^2}}}.2x\,\,\,\,\,\,\,\,\,[\because \dfrac{d}{{dx}}{e^x} = {e^x},\,\dfrac{d}{{dx}}{x^n} = n.{x^{n - 1}}]$

Complete step-by-step solution:
We have to differentiate $\sin \left( {\sin \,{x^2}} \right)$to write $x$.
i.e. $\dfrac{d}{{dx}}\sin (\sin \,{x^2})$
We will use chain rule
i.e. $\dfrac{d}{{dx}}(fog) = \dfrac{{df}}{{dx}}.\dfrac{{dg}}{{dx}}$
$\Rightarrow \dfrac{d}{{dx}}\sin (\sin \,{x^2}) = \dfrac{d}{{dx}}\sin (\sin \,{x^2}).\dfrac{d}{{dx}}(\sin {x^2}).\dfrac{d}{{dx}}{x^2}$
$= \cos (\sin \,{x^2}).\cos \,{x^2}.2x$
Since, $\left[ {\dfrac{d}{{dx}}\sin x = \cos x,\,\dfrac{d}{{dx}}{x^n} = n{x^{n - 1}}} \right]$
So, our required answer is
$\Rightarrow \dfrac{d}{{dx}}\sin (\sin {x^2}) = \cos (\sin {x^2})\cos {x^2}.2x$

Note In such questions we directly differentiate the function i.e. we forget to differentiate inner function or in such we cannot apply the product function rule in this. That means we see such question we think that is very easy and we directly apply $\dfrac{d}{{dx}}(p.q) = \dfrac{{qdP}}{{dx}} + \dfrac{{pdq}}{{dx}}$which leads to a wrong conception.
We use the derivative to determine the maximum and minimum values of particular functions (e.g. cost, strength, amount of material used in a building, profit, loss, etc.). Derivatives are met in many engineering and science problems, especially when modelling the behavior of moving objects.