Let f(x) = begin{cases} sin x, & text{for } x | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) To find $(g \circ f)'(0)$,we must check the left-hand derivative and right-hand derivative at $x = 0$.For the right-hand derivative $(g \circ f)'(

Vedclass · Accepted Answer

None of these

Vedclass · Answer

(D) To find $(g \circ f)'(0)$,we must check the left-hand derivative and right-hand derivative at $x = 0$.For the right-hand derivative $(g \circ f)'(0^+)$:$(g \circ f)(x) = g(f(x)) = g(\sin x) = e^{\sin x}$ for $x \ge 0$.$(g \circ f)'(x) = e^{\sin x} \cdot \cos x$.At $x = 0$,$(g \circ f)'(0^+) = e^{\sin 0} \cdot \cos 0 = e^0 \cdot 1 = 1$.For the left-hand derivative $(g \circ f)'(0^-)$:$(g \circ f)(x) = g(f(x)) = g(1 - \cos x) = e^{1 - \cos x}$ for $x \le 0$.$(g \circ f)'(x) = e^{1 - \cos x} \cdot \sin x$.At $x = 0$,$(g \circ f)'(0^-) = e^{1 - \cos 0} \cdot \sin 0 = e^0 \cdot 0 = 0$.Since the left-hand derivative $(0)$ is not equal to the right-hand derivative $(1)$,the derivative $(g \circ f)'(0)$ does not exist.Therefore,the correct option is $D$.

Let $f(x) = \begin{cases} \sin x, & \text{for } x \ge 0 \\ 1 - \cos x, & \text{for } x \le 0 \end{cases}$ and $g(x) = e^x$. Then $(g \circ f)'(0)$ is

Similar Questions

If the function $f(x) = \begin{cases} \frac{1}{|x|} & , |x| \geq 2 \\ ax^2 + 2b & , |x| < 2 \end{cases}$ is differentiable on $\mathbb{R}$,then $48(a+b)$ is equal to . . . . . . .

If $f(x) = \begin{cases} e^x + ax, & x < 0 \\ b(x - 1)^2, & x \ge 0 \end{cases}$ is differentiable at $x = 0$,then $(a, b)$ is

The function $f(x) = \max \{(1 - x), (1 + x), 2\},$ $x \in ( - \infty , \infty ),$ is

$A$ function $f$ is defined as follows:
$f(x) = \begin{cases} \sin x & \text{if } x \le c \\ ax + b & \text{if } x > c \end{cases}$
where $c$ is a known quantity. If $f$ is derivable at $x = c$,then the values of $a$ and $b$ are . . . . . . and . . . . . . respectively.

If $f(x) = \begin{cases} x + 1, & x < 2 \\ 2x - 1, & x \ge 2 \end{cases}$,then $f'(2)$ equals

Vedclass Test Series

Exam Paper Generator

Online Exam Module