Answer

Verified

440.7k+ views

**Hint**: To solve the above problem we have to know the basic derivatives of \[\sin x\]and \[\dfrac{1}{{{x}^{2}}}\]. After writing the derivatives rewrite the equation with the derivatives of the function.

\[\dfrac{d}{dx}\left( \sin x \right)=\cos x\],\[\dfrac{d}{dx}\left( \dfrac{1}{{{x}^{2}}} \right)=\dfrac{-2}{{{x}^{3}}}\]. We can see one function is inside another we have to find internal derivatives.

**:**

__Complete step-by-step answer__The composite function rule shows us a quicker way. If f(x) = h(g(x)) then f (x) = h (g(x)) × g (x). In words: differentiate the 'outside' function, and then multiply by the derivative of the 'inside' function. ... The composite function rule tells us that f (x) = 17(x2 + 1)16 × 2x.

\[f(x)=\sin \left( \dfrac{1}{{{x}^{2}}} \right)\]. . . . . . . . . . . . . . . . . . . . . (a)

\[\dfrac{d}{dx}\left( \sin x \right)=\cos x\]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

\[\dfrac{d}{dx}\left( \dfrac{1}{{{x}^{2}}} \right)=\dfrac{-2}{{{x}^{3}}}\]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)

Substituting (1) and (2) in (a) we get,

Therefore derivative of the given function is,

\[{{f}^{1}}\left( x \right)=\dfrac{d}{dx}\left( \sin \left( \dfrac{1}{{{x}^{2}}} \right) \right)\]

We know the derivative of \[\sin x\]and \[\dfrac{1}{{{x}^{2}}}\]. By writing the derivatives we get,

Further solving we get the derivative of the function as

\[{{f}^{1}}\left( x \right)=\cos \left( \dfrac{1}{{{x}^{2}}} \right)\left( \dfrac{-2}{{{x}^{3}}} \right)\] . . . . . . . . . . . . . . . . . . . (3)

By solving we get,

\[{{f}^{1}}\left( x \right)=\dfrac{-2}{{{x}^{3}}}\cos \left( \dfrac{1}{{{x}^{2}}} \right)\]

**Note**: In the above problem we have solved the derivative of the trigonometric function. In (3) the formation of \[\dfrac{-2}{{{x}^{3}}}\]is due to function in a function. In this case we have to find an internal derivative. Further solving for \[\dfrac{dy}{dx}\]made us towards a solution. If we are doing derivative means we are finding the slope of a function. Care should be taken while doing calculations.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

A group of fish is known as class 7 english CBSE

The highest dam in India is A Bhakra dam B Tehri dam class 10 social science CBSE

Write all prime numbers between 80 and 100 class 8 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Onam is the main festival of which state A Karnataka class 7 social science CBSE

Who administers the oath of office to the President class 10 social science CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Kolkata port is situated on the banks of river A Ganga class 9 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE