Suppose-function-f-R-R-with-f-2x-3-4x-2-2x-5-and-f-denote-is-derivatif-of-f-function-What-the-value-of-f-2x-3-

Question Number 133616 by benjo_mathlover last updated on 23/Feb/21

Suppose function f : R \to R with

f (2 x - 3) = {4 x}^{2} + 2 x - 5 and f^{'} denote

is derivatif of f function .

What the value of f^{'} (2 x - 3)

Commented by mr W last updated on 23/Feb/21

4 x + 1

Answered by TheSupreme last updated on 23/Feb/21

D (f (g (x))) = g^{'} (x) f^{'} (g (x)) = 8 x + 2

g (x) = 2 x - 3 \to g^{'} (x) = 2

f^{'} (g (x))) = 4 x + 1

Answered by liberty last updated on 23/Feb/21

\Rightarrow f (x) = 4 {(\frac{x + 3}{2})}^{2} + 2 (\frac{x + 3}{2}) - 5

f (x) = x^{2} + 6 x + 9 + x + 3 - 5

f (x) = x^{2} + 7 x + 7

\Rightarrow f^{'} (x) = 2 x + 7

f^{'} (2 x - 3) = 2 (2 x - 3) + 7 = 4 x + 1

Answered by mr W last updated on 23/Feb/21

l e t u = 2 x - 3

\Rightarrow \frac{d x}{d u} = \frac{1}{2}

f (u) = {4 x}^{2} + 2 x - 5

f^{'} (u) = \frac{d f}{d u} = \frac{d f}{d x} \times \frac{d x}{d u} = (8 x + 2) \frac{1}{2} = 4 x + 1

i . e . f^{'} (2 x - 3) = 4 x + 1

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