Answer
425.4k+ views
Hint: First, we can see clearly that the function here mentioned is a composite function which is the combination of several functions. Then, we need to use the chain rule to find the derivative of such types of functions. Then, we can clearly see that by chain rule the continuous differentiation of the function continues till it reaches the value of differentiation of x which is 1 to get the desired answer.
Complete step-by-step answer:
In this question, we are supposed to find the differentiation of the function $ y=f\left[ \phi \left( \psi \left( h\left( g\left( x \right) \right) \right) \right) \right] $ .
Now, we can see clearly that the function here mentioned is a composite function which is the combination of several functions.
So, we need to use the chain rule to find the derivative of such type of functions.
Now, before proceeding for the actual function let the function b y=f(g(x)) and its differentiation by chain rule is given by:
$ \begin{align}
& \dfrac{dy}{dx}={f}'\left( g\left( x \right) \right)\centerdot \dfrac{d}{dx}g\left( x \right) \\
& \Rightarrow \dfrac{dy}{dx}={f}'\left( g\left( x \right) \right)\centerdot {g}'\left( x \right)\centerdot \dfrac{d}{dx}x \\
& \Rightarrow \dfrac{dy}{dx}={f}'\left( g\left( x \right) \right)\centerdot {g}'\left( x \right)\centerdot 1 \\
\end{align} $
So, we can clearly that by chain rule the continuous differentiation of the function continues till it reaches the value of differentiation of x which is 1.
Now, by applying the same rule to the given function in the question as $ y=f\left[ \phi \left( \psi \left( h\left( g\left( x \right) \right) \right) \right) \right] $ and solving it by chain rule, we get:
$ \begin{align}
& \dfrac{dy}{dx}={f}'\left( \phi \left( \psi \left( h\left( g\left( x \right) \right) \right) \right) \right)\centerdot \dfrac{d}{dx}\left[ \phi \left( \psi \left( h\left( g\left( x \right) \right) \right) \right) \right] \\
& \Rightarrow \dfrac{dy}{dx}={f}'\left( \phi \left( \psi \left( h\left( g\left( x \right) \right) \right) \right) \right)\centerdot {\phi }'\left( \psi \left( h\left( g\left( x \right) \right) \right) \right)\dfrac{d}{dx}\left[ \psi \left( h\left( g\left( x \right) \right) \right) \right] \\
& \Rightarrow \dfrac{dy}{dx}={f}'\left( \phi \left( \psi \left( h\left( g\left( x \right) \right) \right) \right) \right)\centerdot {\phi }'\left( \psi \left( h\left( g\left( x \right) \right) \right) \right)\centerdot {\psi }'\left( h\left( g\left( x \right) \right) \right)\dfrac{d}{dx}\left[ h\left( g\left( x \right) \right) \right] \\
& \Rightarrow \dfrac{dy}{dx}={f}'\left( \phi \left( \psi \left( h\left( g\left( x \right) \right) \right) \right) \right)\centerdot {\phi }'\left( \psi \left( h\left( g\left( x \right) \right) \right) \right)\centerdot {\psi }'\left( h\left( g\left( x \right) \right) \right)\centerdot {h}'\left( g\left( x \right) \right)\dfrac{d}{dx}\left[ g\left( x \right) \right] \\
& \Rightarrow \dfrac{dy}{dx}={f}'\left( \phi \left( \psi \left( h\left( g\left( x \right) \right) \right) \right) \right)\centerdot {\phi }'\left( \psi \left( h\left( g\left( x \right) \right) \right) \right)\centerdot {\psi }'\left( h\left( g\left( x \right) \right) \right)\centerdot {h}'\left( g\left( x \right) \right)\centerdot {g}'\left( x \right)\centerdot \dfrac{d}{dx}x \\
& \Rightarrow \dfrac{dy}{dx}={f}'\left( \phi \left( \psi \left( h\left( g\left( x \right) \right) \right) \right) \right)\centerdot {\phi }'\left( \psi \left( h\left( g\left( x \right) \right) \right) \right)\centerdot {\psi }'\left( h\left( g\left( x \right) \right) \right)\centerdot {h}'\left( g\left( x \right) \right)\centerdot {g}'\left( x \right) \\
\end{align} $
So, the differentiation of the function $ y=f\left[ \phi \left( \psi \left( h\left( g\left( x \right) \right) \right) \right) \right] $ is $ {f}'\left( \phi \left( \psi \left( h\left( g\left( x \right) \right) \right) \right) \right)\centerdot {\phi }'\left( \psi \left( h\left( g\left( x \right) \right) \right) \right)\centerdot {\psi }'\left( h\left( g\left( x \right) \right) \right)\centerdot {h}'\left( g\left( x \right) \right)\centerdot {g}'\left( x \right) $ .
Now, the answer above found can be written as $ {f}'\left( \phi o\psi ohog \right)\centerdot {\phi }'\left( \psi ohog \right)\centerdot {h}'\left( g\left( x \right) \right)\centerdot {g}'\left( x \right) $ .
So, the correct answer is “Option A”.
Note: Now, the only mistake we can make in these types of questions is that we don’t apply the chain rule and just solve till the first bracket and give that as an answer. So, if we do the same with this question our answer will be as:
$ \dfrac{dy}{dx}={f}'\left( \phi \left( \psi \left( h\left( g\left( x \right) \right) \right) \right) \right) $ And we give it an answer which doesn’t match with any option and we will mark one of these answer which is the wrong option. So, be careful while performing differentiation of the composite functions.
Complete step-by-step answer:
In this question, we are supposed to find the differentiation of the function $ y=f\left[ \phi \left( \psi \left( h\left( g\left( x \right) \right) \right) \right) \right] $ .
Now, we can see clearly that the function here mentioned is a composite function which is the combination of several functions.
So, we need to use the chain rule to find the derivative of such type of functions.
Now, before proceeding for the actual function let the function b y=f(g(x)) and its differentiation by chain rule is given by:
$ \begin{align}
& \dfrac{dy}{dx}={f}'\left( g\left( x \right) \right)\centerdot \dfrac{d}{dx}g\left( x \right) \\
& \Rightarrow \dfrac{dy}{dx}={f}'\left( g\left( x \right) \right)\centerdot {g}'\left( x \right)\centerdot \dfrac{d}{dx}x \\
& \Rightarrow \dfrac{dy}{dx}={f}'\left( g\left( x \right) \right)\centerdot {g}'\left( x \right)\centerdot 1 \\
\end{align} $
So, we can clearly that by chain rule the continuous differentiation of the function continues till it reaches the value of differentiation of x which is 1.
Now, by applying the same rule to the given function in the question as $ y=f\left[ \phi \left( \psi \left( h\left( g\left( x \right) \right) \right) \right) \right] $ and solving it by chain rule, we get:
$ \begin{align}
& \dfrac{dy}{dx}={f}'\left( \phi \left( \psi \left( h\left( g\left( x \right) \right) \right) \right) \right)\centerdot \dfrac{d}{dx}\left[ \phi \left( \psi \left( h\left( g\left( x \right) \right) \right) \right) \right] \\
& \Rightarrow \dfrac{dy}{dx}={f}'\left( \phi \left( \psi \left( h\left( g\left( x \right) \right) \right) \right) \right)\centerdot {\phi }'\left( \psi \left( h\left( g\left( x \right) \right) \right) \right)\dfrac{d}{dx}\left[ \psi \left( h\left( g\left( x \right) \right) \right) \right] \\
& \Rightarrow \dfrac{dy}{dx}={f}'\left( \phi \left( \psi \left( h\left( g\left( x \right) \right) \right) \right) \right)\centerdot {\phi }'\left( \psi \left( h\left( g\left( x \right) \right) \right) \right)\centerdot {\psi }'\left( h\left( g\left( x \right) \right) \right)\dfrac{d}{dx}\left[ h\left( g\left( x \right) \right) \right] \\
& \Rightarrow \dfrac{dy}{dx}={f}'\left( \phi \left( \psi \left( h\left( g\left( x \right) \right) \right) \right) \right)\centerdot {\phi }'\left( \psi \left( h\left( g\left( x \right) \right) \right) \right)\centerdot {\psi }'\left( h\left( g\left( x \right) \right) \right)\centerdot {h}'\left( g\left( x \right) \right)\dfrac{d}{dx}\left[ g\left( x \right) \right] \\
& \Rightarrow \dfrac{dy}{dx}={f}'\left( \phi \left( \psi \left( h\left( g\left( x \right) \right) \right) \right) \right)\centerdot {\phi }'\left( \psi \left( h\left( g\left( x \right) \right) \right) \right)\centerdot {\psi }'\left( h\left( g\left( x \right) \right) \right)\centerdot {h}'\left( g\left( x \right) \right)\centerdot {g}'\left( x \right)\centerdot \dfrac{d}{dx}x \\
& \Rightarrow \dfrac{dy}{dx}={f}'\left( \phi \left( \psi \left( h\left( g\left( x \right) \right) \right) \right) \right)\centerdot {\phi }'\left( \psi \left( h\left( g\left( x \right) \right) \right) \right)\centerdot {\psi }'\left( h\left( g\left( x \right) \right) \right)\centerdot {h}'\left( g\left( x \right) \right)\centerdot {g}'\left( x \right) \\
\end{align} $
So, the differentiation of the function $ y=f\left[ \phi \left( \psi \left( h\left( g\left( x \right) \right) \right) \right) \right] $ is $ {f}'\left( \phi \left( \psi \left( h\left( g\left( x \right) \right) \right) \right) \right)\centerdot {\phi }'\left( \psi \left( h\left( g\left( x \right) \right) \right) \right)\centerdot {\psi }'\left( h\left( g\left( x \right) \right) \right)\centerdot {h}'\left( g\left( x \right) \right)\centerdot {g}'\left( x \right) $ .
Now, the answer above found can be written as $ {f}'\left( \phi o\psi ohog \right)\centerdot {\phi }'\left( \psi ohog \right)\centerdot {h}'\left( g\left( x \right) \right)\centerdot {g}'\left( x \right) $ .
So, the correct answer is “Option A”.
Note: Now, the only mistake we can make in these types of questions is that we don’t apply the chain rule and just solve till the first bracket and give that as an answer. So, if we do the same with this question our answer will be as:
$ \dfrac{dy}{dx}={f}'\left( \phi \left( \psi \left( h\left( g\left( x \right) \right) \right) \right) \right) $ And we give it an answer which doesn’t match with any option and we will mark one of these answer which is the wrong option. So, be careful while performing differentiation of the composite functions.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)