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Question: How do you differentiate \( f(t) = {\sin ^2}({e^{{{\sin }^2}t}}) \) using the chain rule?...

How do you differentiate f(t)=sin⁡2(esin⁡2t)f(t) = {\sin ^2}({e^{{{\sin }^2}t}}) using the chain rule?

Explanation

Solution

Hint : In the method of differentiating by chain rule we differentiate the outer function then in the successive steps we differentiate the inside functions until we reach the variable which we are differentiating. Thus for the above question first we differentiate the outer function which is given as sin⁡\sin but instead of the common variable tt another function is instead present so we will then differentiate that inside function as well and keep on repeating the chain until we reach our basic variable which here is tt .

Complete step by step solution:
To solve the above function first we solve the outer function then we solve the functions present inside the outer function and keep on repeating it until we reach the basic variable itself.
So for the first step we differentiate the outer function which in this given case is sin⁡2t{\sin ^2}t
Given y=sin⁡2(esin⁡2t)y = {\sin ^2}({e^{{{\sin }^2}t}})
dydt=cos2t (esin⁡2t).ddtesin⁡2t\dfrac{{dy}}{{dt}} = co{s^2}t{\text{ }}({e^{{{\sin }^2}t}}).\dfrac{d}{{dt}}{e^{{{\sin }^2}t}}
Upon further solving the differentiation of the inside function we get
dydt=cos2t (esin⁡2t).esin⁡2t.ddtsin⁡2t\dfrac{{dy}}{{dt}} = co{s^2}t{\text{ }}({e^{{{\sin }^2}t}}).{e^{{{\sin }^2}t}}.\dfrac{d}{{dt}}{\sin ^2}t
And at last solving the innermost function we get the final answer as
dydt=cos2t (esin⁡2t).esin⁡2t.2cos⁡tsint\dfrac{{dy}}{{dt}} = co{s^2}t{\text{ }}({e^{{{\sin }^2}t}}).{e^{{{\sin }^2}t}}.{2\cos}tsint
Thus the following differentiation is solved. This had three functions nested inside each other but we differentiated every function one at a time until all the given functions were differentiated.
So, the correct answer is “ dydt=cos2t (esin⁡2t).esin⁡2t.2cos⁡tsint\dfrac{{dy}}{{dt}} = co{s^2}t{\text{ }}({e^{{{\sin }^2}t}}).{e^{{{\sin }^2}t}}.{2\cos}tsint ”.

Note : For the differentiation using chain rule the differentiation must be done with every function beginning from the outer function to the inner function taking one function at a time. Each function given here has been differentiated by using the standard formula. The above used formula are:
ddx(sin⁡x)=cos⁡x\dfrac{d}{{dx}}(\sin x) = \cos x
ddx(ex)=ex\dfrac{d}{{dx}}({e^x}) = {e^x}