Advertisements
Advertisements
प्रश्न
If f (1) = 1, f' (1) = 2, then write the value of \[\lim_{x \to 1} \frac{\sqrt{f (x)} - 1}{\sqrt{x} - 1}\]
Advertisements
उत्तर
\[\lim_{x \to 1} \frac{\sqrt{f\left( x \right)} - 1}{\sqrt{x} - 1}\]
\[ = \lim_{x \to 1} \frac{\sqrt{f\left( x \right)} - 1}{\sqrt{x} - 1} \times \frac{\sqrt{f\left( x \right)} + 1}{\sqrt{f\left( x \right)} + 1} \times \frac{\sqrt{x} + 1}{\sqrt{x} + 1}\]
\[ = \lim_{x \to 1} \frac{\left( f\left( x \right) - 1 \right)\left( \sqrt{x} + 1 \right)}{\left( x - 1 \right)\left( \sqrt{f\left( x \right)} + 1 \right)}\]
\[ = \lim_{x \to 1} \frac{f\left( x \right) - 1}{x - 1} \times \lim_{x \to 1} \frac{\left( \sqrt{x} + 1 \right)}{\left( \sqrt{f\left( x \right)} + 1 \right)}\]
\[ = f'\left( 1 \right) \times \frac{1 + 1}{\sqrt{f\left( 1 \right)} + 1}\]
\[ = 2 \times \frac{2}{1 + 1}\]
\[ = 2\]
APPEARS IN
संबंधित प्रश्न
Find the derivative of x2 – 2 at x = 10.
Find the derivative of (5x3 + 3x – 1) (x – 1).
Find the derivative of x–4 (3 – 4x–5).
Find the derivative of `2/(x + 1) - x^2/(3x -1)`.
Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):
`(px+ q) (r/s + s)`
Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):
`(ax + b)/(px^2 + qx + r)`
Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):
(ax + b)n (cx + d)m
Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):
`(sin x + cos x)/(sin x - cos x)`
Find the derivative of the following function at the indicated point:
sin x at x =\[\frac{\pi}{2}\]
\[\frac{x + 1}{x + 2}\]
Differentiate of the following from first principle:
eax + b
Differentiate each of the following from first principle:
\[\frac{\sin x}{x}\]
Differentiate each of the following from first principle:
\[\frac{\cos x}{x}\]
Differentiate each of the following from first principle:
\[\sqrt{\sin (3x + 1)}\]
Differentiate each of the following from first principle:
\[e^\sqrt{2x}\]
Differentiate each of the following from first principle:
\[a^\sqrt{x}\]
tan (2x + 1)
x4 − 2 sin x + 3 cos x
3x + x3 + 33
ex log a + ea long x + ea log a
(2x2 + 1) (3x + 2)
log3 x + 3 loge x + 2 tan x
\[\left( x + \frac{1}{x} \right)\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]
\[\frac{(x + 5)(2 x^2 - 1)}{x}\]
\[\text{ If } y = \left( \frac{2 - 3 \cos x}{\sin x} \right), \text{ find } \frac{dy}{dx} at x = \frac{\pi}{4}\]
For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]
xn loga x
x5 ex + x6 log x
(x sin x + cos x ) (ex + x2 log x)
sin2 x
Differentiate in two ways, using product rule and otherwise, the function (1 + 2 tan x) (5 + 4 cos x). Verify that the answers are the same.
(ax + b)n (cx + d)n
\[\frac{x^2 + 1}{x + 1}\]
\[\frac{e^x - \tan x}{\cot x - x^n}\]
\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\]
Write the value of \[\lim_{x \to c} \frac{f(x) - f(c)}{x - c}\]
Mark the correct alternative in each of the following:
If\[f\left( x \right) = \frac{x^n - a^n}{x - a}\] then \[f'\left( a \right)\]
