Advertisements
Advertisements
Question
tan2 x
Advertisements
Solution
\[\frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\tan^2 \left( x + h \right) - \tan^2 x}{h}\]
\[ = \lim_{h \to 0} \frac{\left[ \tan \left( x + h \right) + \tan x \right]\left[ \tan \left( x + h \right) - \tan x \right]}{h}\]
\[ = \lim_{h \to 0} \frac{\left[ \frac{\sin \left( x + h \right)}{\cos \left( x + h \right)} + \frac{\sin x}{\cos x} \right]\left[ \frac{\sin (x + h)}{\cos (x + h)} - \frac{\sin x}{\cos x} \right]}{h}\]
\[ = \lim_{h \to 0} \frac{\left[ \sin \left( x + h \right) \cos x + \cos \left( x + h \right) \sin x \right]\left[ \sin \left( x + h \right) \cos x - \cos \left( x + h \right) \sin x \right]}{h \cos^2 x \cos^2 \left( x + h \right)}\]
\[ = \lim_{h \to 0} \frac{\left[ \sin \left( 2x + h \right) \right]\left[ \sin h \right]}{h \cos^2 x \cos^2 \left( x + h \right)}\]
\[ = \frac{1}{\cos^2 x} \lim_{h \to 0} \sin \left( 2x + h \right) \lim_{h \to 0} \frac{\sin h}{h} \lim_{h \to 0} \frac{1}{\cos^2 \left( x + h \right)}\]
\[ = \frac{1}{\cos^2 x} \sin \left( 2x \right) \left( 1 \right)\frac{1}{\cos^2 x}\]
\[ = \frac{1}{\cos^2 x} 2 \sin x \cos x \frac{1}{\cos^2 x}\]
\[ = 2 \times \frac{\sin x}{\cos x} \times \frac{1}{\cos^2 x}\]
\[ = 2 \tan x \sec^2 x\]
APPEARS IN
RELATED QUESTIONS
Find the derivative of 99x at x = 100.
Find the derivative of (5x3 + 3x – 1) (x – 1).
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):
`a/x^4 = b/x^2 + cos x`
Find the derivative of f (x) x at x = 1
Find the derivative of f (x) = tan x at x = 0
\[\frac{2}{x}\]
(x2 + 1) (x − 5)
x ex
Differentiate each of the following from first principle:
\[\sqrt{\sin 2x}\]
Differentiate each of the following from first principle:
\[e^\sqrt{ax + b}\]
tan (2x + 1)
\[\frac{x^3}{3} - 2\sqrt{x} + \frac{5}{x^2}\]
\[\frac{( x^3 + 1)(x - 2)}{x^2}\]
\[\frac{a \cos x + b \sin x + c}{\sin x}\]
If for f (x) = λ x2 + μ x + 12, f' (4) = 15 and f' (2) = 11, then find λ and μ.
For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]
logx2 x
x4 (5 sin x − 3 cos x)
x5 (3 − 6x−9)
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.
(3x2 + 2)2
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.
(x + 2) (x + 3)
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)
\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\]
\[\frac{{10}^x}{\sin x}\]
\[\frac{3^x}{x + \tan x}\]
\[\frac{x}{1 + \tan x}\]
\[\frac{p x^2 + qx + r}{ax + b}\]
\[\frac{\sec x - 1}{\sec x + 1}\]
\[\frac{x + \cos x}{\tan x}\]
\[\frac{x}{\sin^n x}\]
Mark the correct alternative in of the following:
If \[f\left( x \right) = \frac{x - 4}{2\sqrt{x}}\]
Mark the correct alternative in of the following:
If\[y = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + . . .\]then \[\frac{dy}{dx} =\]
Mark the correct alternative in each of the following:
If\[y = \frac{\sin x + \cos x}{\sin x - \cos x}\] then \[\frac{dy}{dx}\]at x = 0 is
Find the derivative of 2x4 + x.
Find the derivative of x2 cosx.
Find the derivative of f(x) = tan(ax + b), by first principle.
