Advertisements
Advertisements
प्रश्न
\[\frac{x + 2}{3x + 5}\]
Advertisements
उत्तर
\[ \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\frac{x + h + 2}{3\left( x + h \right) + 5} - \frac{x + 2}{3x + 5}}{h}\]
\[ = \lim_{h \to 0} \frac{\frac{x + h + 2}{3x + 3h + 5} - \frac{x + 2}{3x + 5}}{h}\]
\[ = \lim_{h \to 0} \frac{\left( x + h + 2 \right)\left( 3x + 5 \right) - \left( 3x + 3h + 5 \right)\left( x + 2 \right)}{h\left( 3x + 3h + 5 \right)\left( 3x + 5 \right)}\]
\[ = \lim_{h \to 0} \frac{3 x^2 + 3xh + 6x + 5x + 5h + 10 - 3 x^2 - 3xh - 5x - 6x - 6h - 10}{h\left( 3x + 3h + 5 \right)\left( 3x + 5 \right)}\]
\[ = \lim_{h \to 0} \frac{- h}{h\left( 3x + 3h + 5 \right)\left( 3x + 5 \right)}\]
\[ = \lim_{h \to 0} \frac{- 1}{\left( 3x + 3h + 5 \right)\left( 3x + 5 \right)}\]
\[ = \frac{- 1}{\left( 3x + 5 \right)^2}\]
APPEARS IN
संबंधित प्रश्न
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) (cx + d)2
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):
`(1 + 1/x)/(1- 1/x)`
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):
`a/x^4 = b/x^2 + cos x`
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):
`4sqrtx - 2`
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):
`(sec x - 1)/(sec x + 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):
`(a + bsin x)/(c + dcosx)`
Find the derivative of f (x) = x2 − 2 at x = 10
Find the derivative of f (x) = cos x at x = 0
\[\frac{1}{\sqrt{x}}\]
x2 + x + 3
(x2 + 1) (x − 5)
Differentiate of the following from first principle:
eax + b
Differentiate each of the following from first principle:
\[\sqrt{\sin 2x}\]
Differentiate each of the following from first principle:
sin x + cos x
Differentiate each of the following from first principle:
x2 ex
Differentiate each of the following from first principle:
\[3^{x^2}\]
\[\frac{2 x^2 + 3x + 4}{x}\]
cos (x + a)
\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]
If for f (x) = λ x2 + μ x + 12, f' (4) = 15 and f' (2) = 11, then find λ and μ.
xn loga x
(1 +x2) cos x
Differentiate each of the following functions by the product rule and the other method and verify that answer from both the methods is the same.
(3 sec x − 4 cosec x) (−2 sin x + 5 cos x)
\[\frac{x^2 + 1}{x + 1}\]
\[\frac{e^x}{1 + x^2}\]
\[\frac{x \sin x}{1 + \cos x}\]
\[\frac{\sin x - x \cos x}{x \sin x + \cos x}\]
\[\frac{1 + \log x}{1 - \log x}\]
\[\frac{x + \cos x}{\tan x}\]
\[\frac{ax + b}{p x^2 + qx + r}\]
Write the value of \[\lim_{x \to c} \frac{f(x) - f(c)}{x - c}\]
Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.
If f (x) = \[\frac{x^2}{\left| x \right|},\text{ write }\frac{d}{dx}\left( f (x) \right)\]
If f (x) = \[\log_{x_2}\]write the value of f' (x).
Mark the correct alternative in of the following:
If \[y = \frac{1 + \frac{1}{x^2}}{1 - \frac{1}{x^2}}\] then \[\frac{dy}{dx} =\]
Mark the correct alternative in of the following:
If \[f\left( x \right) = x^{100} + x^{99} + . . . + x + 1\] then \[f'\left( 1 \right)\] is equal to
