Let f be a differentiable function such that f(1) = 2 and f '(x) = f(x) for all x ∈ R. If h(x) = f(f(x)), then h'(1) is equal to :

Question

Let f be a differentiable function such that f(1) = 2 and f '(x) = f(x) for all x ∈ R. If h(x) = f(f(x)), then h'(1) is equal to : (A) 4e (B) 4e2 (C) 2e (D) 2e2

Tardigrade Admin · Accepted Answer

(f'(x)/f(x)) = 1 ∀ x ∈ R Intergrate use f(1) = 2 f(x) = 2ex-1 ⇒ f'(x) = 2ex-1 h(x) = f(f(x)) ⇒ h'x) = f'(f(x)) f'(x) h'(1) = f'(f(f)) f'(1) = f'(2) f'(1) =2e .2 =4e

Q. Let $f$ be a differentiable function such that $f (1) = 2$ and $f^{'} (x) = f (x)$ for all $x \in R$ . If $h (x) = f (f (x))$ , then $h^{'} (1)$ is equal to :

4e

4e²

2e

2e²

$\frac{f ^{'} ( x )}{f ( x )} = 1\forall x \in R$
Intergrate & use $f (1) = 2$
f(x) = 2e $^{x - 1} \Rightarrow f^{'} (x) = 2 e^{x - 1}$
$h (x) = f (f (x)) \Rightarrow h^{'} x) = f^{'} (f (x)) f^{'} (x)$
$h^{'} (1) = f^{'} (f (f)) f^{'} (1)$
$= f^{'} (2) f^{'} (1)$
$= 2 e .2 = 4 e$

Q. Let f be a differentiable function such that f(1)=2 and f′(x)=f(x) for all x∈R. If h(x)=f(f(x)), then h′(1) is equal to :

4e

4e2

2e

2e2

f(x)f′(x)​=1∀x∈R Intergrate & use f(1)=2 f(x) = 2ex−1⇒f′(x)=2ex−1 h(x)=f(f(x))⇒h′x)=f′(f(x))f′(x) h′(1)=f′(f(f))f′(1) =f′(2)f′(1) =2e.2=4e

Q. Let $f$ be a differentiable function such that $f (1) = 2$ and $f^{'} (x) = f (x)$ for all $x \in R$ . If $h (x) = f (f (x))$ , then $h^{'} (1)$ is equal to :

4e²

2e²

$\frac{f ^{'} ( x )}{f ( x )} = 1\forall x \in R$
Intergrate & use $f (1) = 2$
f(x) = 2e $^{x - 1} \Rightarrow f^{'} (x) = 2 e^{x - 1}$
$h (x) = f (f (x)) \Rightarrow h^{'} x) = f^{'} (f (x)) f^{'} (x)$
$h^{'} (1) = f^{'} (f (f)) f^{'} (1)$
$= f^{'} (2) f^{'} (1)$
$= 2 e .2 = 4 e$