Question

3.
If f(x)= g(u) and u = u(x) then
(a) f'(x)=g'(u)
(b)f'(x) =g '(u). u '(x)
(c) f'(x)=u'(x)
(d) none of these​

Answer 1

If f(x) = g(u) and u = u(x) then f'(x) = g'(u) . u'(x)

Given :

f(x) = g(u) and u = u(x)

To find :

The value of f'(x) is

(a) f'(x) = g'(u)

(b) f'(x) = g'(u).u'(x)

(c) f'(x) = u'(x)

(d) none of these

Solution :

Step 1 of 2 :

Write down the given function

Here the given function is

f(x) = g(u) and u = u(x)

Step 2 of 2 :

Find the value of f'(x)

$\displaystyle \sf{f'(x) }$

$\displaystyle \sf{ = \frac{d}{dx} \bigg( f(x)\bigg) }$

$\displaystyle \sf{ = \frac{d}{dx} \bigg(g(u)\bigg) }\: \: \: \bigg[ \: \because \: f(x) = g(u)\bigg]$

$\displaystyle \sf{ = \frac{d}{du} \bigg(g(u)\bigg) }. \frac{du}{dx}$

$\displaystyle \sf{ = g'(u).u'(x)}$

Hence the correct option is (b) f'(x) = g'(u).u'(x)

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Answer 2

Answer:

The correct option is (b) f'(x) = g'(u).u'(x)

Given :

f(x) = g(u) and u = u(x)

To find :

The value of f'(x) is

Solution :

Step 1 of 2 :

The formal chain rule is as follows.  When a function takes the following form:    

$y=f(u) \text { and } u=g(x) \text { such that } y=f[g(x)]$

Then the derivative of y with respect to x is defined as:   $\frac{d y}{d x}=\frac{d y}{d u} \cdot \frac{d u}{d x}$            

Write down the given function

Here the given function is

f(x) = g(u) and u = u(x)

Step 2 of 2 :

Find the value of f'(x)

$\begin{aligned}&f^{\prime}(x) \\&=\frac{d}{d x}(f(x)) \\&=\frac{d}{d x}(g(u))[\because f(x)=g(u)] \\&=\frac{d}{d u}(g(u)) \cdot \frac{d u}{d x} \\&=g^{\prime}(u) \cdot u^{\prime}(x)\end{aligned}$

Hence the correct option is (b) f'(x) = g'(u).u'(x)