Let f ( x ) be a polynomial in x. Then the second derivative of f(ex) w.r.t. x is

Question

Let f ( x ) be a polynomial in x. Then the second derivative of f(ex) w.r.t. x is (A) f prime prime(ex) ⋅ ex+f prime(ex) (B) f prime prime(ex) ⋅ e2 x+f prime(ex) ⋅ e2 x (C) f prime prime(ex) e2 x (D) f prime prime(ex) ⋅ e2 x+f prime(ex) ⋅ ex

Tardigrade Admin · Accepted Answer

(d/d x)(f(ex))=f prime(ex) ⋅ ex (d2/d x2)(f(ex))=ex f prime(ex)+e2 x f prime prime(ex)

Q. Let $f (x)$ be a polynomial in $x$ . Then the second derivative of $f (e^{x})$ w.r.t. $x$ is

$f^{''} (e^{x}) \cdot e^{x} + f^{'} (e^{x})$

$f^{''} (e^{x}) \cdot e^{2 x} + f^{'} (e^{x}) \cdot e^{2 x}$

$f^{''} (e^{x}) e^{2 x}$

$f^{''} (e^{x}) \cdot e^{2 x} + f^{'} (e^{x}) \cdot e^{x}$

$\frac{d}{d x} (f (e^{x})) = f^{'} (e^{x}) \cdot e^{x}$
$\frac{d ^{2}}{d x ^{2}} (f (e^{x})) = e^{x} f^{'} (e^{x}) + e^{2 x} f^{''} (e^{x})$

Q. Let f(x) be a polynomial in x. Then the second derivative of f(ex) w.r.t. x is

f′′(ex)⋅ex+f′(ex)

f′′(ex)⋅e2x+f′(ex)⋅e2x

f′′(ex)e2x

f′′(ex)⋅e2x+f′(ex)⋅ex