Advertisements
Advertisements
प्रश्न
tan2 x
Advertisements
उत्तर
\[\frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\tan^2 \left( x + h \right) - \tan^2 x}{h}\]
\[ = \lim_{h \to 0} \frac{\left[ \tan \left( x + h \right) + \tan x \right]\left[ \tan \left( x + h \right) - \tan x \right]}{h}\]
\[ = \lim_{h \to 0} \frac{\left[ \frac{\sin \left( x + h \right)}{\cos \left( x + h \right)} + \frac{\sin x}{\cos x} \right]\left[ \frac{\sin (x + h)}{\cos (x + h)} - \frac{\sin x}{\cos x} \right]}{h}\]
\[ = \lim_{h \to 0} \frac{\left[ \sin \left( x + h \right) \cos x + \cos \left( x + h \right) \sin x \right]\left[ \sin \left( x + h \right) \cos x - \cos \left( x + h \right) \sin x \right]}{h \cos^2 x \cos^2 \left( x + h \right)}\]
\[ = \lim_{h \to 0} \frac{\left[ \sin \left( 2x + h \right) \right]\left[ \sin h \right]}{h \cos^2 x \cos^2 \left( x + h \right)}\]
\[ = \frac{1}{\cos^2 x} \lim_{h \to 0} \sin \left( 2x + h \right) \lim_{h \to 0} \frac{\sin h}{h} \lim_{h \to 0} \frac{1}{\cos^2 \left( x + h \right)}\]
\[ = \frac{1}{\cos^2 x} \sin \left( 2x \right) \left( 1 \right)\frac{1}{\cos^2 x}\]
\[ = \frac{1}{\cos^2 x} 2 \sin x \cos x \frac{1}{\cos^2 x}\]
\[ = 2 \times \frac{\sin x}{\cos x} \times \frac{1}{\cos^2 x}\]
\[ = 2 \tan x \sec^2 x\]
APPEARS IN
संबंधित प्रश्न
Find the derivative of 99x at x = 100.
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):
(x + a)
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):
`1/(ax^2 + bx + c)`
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):
`(a + bsin x)/(c + dcosx)`
Find the derivative of f (x) = cos x at x = 0
\[\frac{1}{x^3}\]
\[\frac{x^2 - 1}{x}\]
\[\frac{x + 2}{3x + 5}\]
\[\sqrt{2 x^2 + 1}\]
Differentiate of the following from first principle:
\[\cos\left( x - \frac{\pi}{8} \right)\]
Differentiate of the following from first principle:
sin (2x − 3)
Differentiate each of the following from first principle:
\[\frac{\sin x}{x}\]
Differentiate each of the following from first principle:
sin x + cos x
\[\sqrt{\tan x}\]
\[\tan \sqrt{x}\]
\[\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^3\]
x2 ex log x
xn tan x
x3 ex cos x
x−4 (3 − 4x−5)
x−3 (5 + 3x)
(ax + b) (a + d)2
(ax + b)n (cx + d)n
\[\frac{x^2 + 1}{x + 1}\]
\[\frac{2x - 1}{x^2 + 1}\]
\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\]
\[\frac{2^x \cot x}{\sqrt{x}}\]
\[\frac{x^2 - x + 1}{x^2 + x + 1}\]
\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\]
\[\frac{{10}^x}{\sin x}\]
If f (x) = |x| + |x−1|, write the value of \[\frac{d}{dx}\left( f (x) \right)\]
Write the derivative of f (x) = 3 |2 + x| at x = −3.
Mark the correct alternative in of the following:
If\[f\left( x \right) = 1 - x + x^2 - x^3 + . . . - x^{99} + x^{100}\]then \[f'\left( 1 \right)\]
Find the derivative of 2x4 + x.
`(a + b sin x)/(c + d cos x)`
