Advertisements
Advertisements
Question
\[\tan \sqrt{x}\]
Advertisements
Solution
\[Let f(x) = \tan\sqrt{x}\]
\[\text{ Thus, we have }: \]
\[(x + h) = \tan\sqrt{x + h}\]
\[\frac{d}{dx}(f(x)) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\]
\[ = \lim_{h \to 0} \frac{\tan\sqrt{x + h} - \tan\sqrt{x}}{h}\]
\[ = \lim_{h \to 0} \frac{\sin \left( \sqrt{x + h} - \sqrt{x} \right)}{h \cos\sqrt{x + h} \cos \sqrt{x}} \left[ \because \tan A - \tan B = \frac{\sin(A - B)}{\cos A \cos B} \right] \]
\[ = \lim_{h \to 0} \frac{\sin \left( \sqrt{x + h} - \sqrt{x} \right)}{\left( x + h - x \right) \cos\sqrt{x + h} \cos \sqrt{x}} \]
\[ = \lim_{h \to 0} \frac{\sin \left( \sqrt{x + h} - \sqrt{x} \right)}{\left( \sqrt{x + h} - \sqrt{x} \right)\left( \sqrt{x + h} - \sqrt{x} \right)\cos\sqrt{x + h} \cos \sqrt{x}}\]
\[ = \lim_{h \to 0} \frac{\sin \left( \sqrt{x + h} - \sqrt{x} \right)}{\left( \sqrt{x + h} - \sqrt{x} \right)} . \lim_{h \to 0} \frac{1}{\left( \sqrt{x + h} + \sqrt{x} \right)\cos\sqrt{x + h}\cos\sqrt{x}} \left[ \because \lim_{h \to 0} \frac{\sin\left( \sqrt{x + h} - \sqrt{x} \right)}{\sqrt{x + h} - \sqrt{x}} = 1 \right]\]
\[ = 1 \times \frac{1}{2\sqrt{x}\cos\sqrt{x}\cos\sqrt{x}}\]
\[ = \frac{1}{2\sqrt{x}} \sec^2 \sqrt{x}\]
APPEARS IN
RELATED QUESTIONS
Find the derivative of x2 – 2 at x = 10.
Find the derivative of `2x - 3/4`
Find the derivative of x5 (3 – 6x–9).
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):
`(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):
sinn x
\[\frac{x^2 - 1}{x}\]
(x + 2)3
(x2 + 1) (x − 5)
\[\frac{2x + 3}{x - 2}\]
x ex
Differentiate of the following from first principle:
x sin x
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:
\[\sqrt{\sin (3x + 1)}\]
\[\sqrt{\tan x}\]
\[\left( x + \frac{1}{x} \right)\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]
\[\frac{2 x^2 + 3x + 4}{x}\]
\[\frac{( x^3 + 1)(x - 2)}{x^2}\]
\[\text{ If } y = \frac{2 x^9}{3} - \frac{5}{7} x^7 + 6 x^3 - x, \text{ find } \frac{dy}{dx} at x = 1 .\]
If for f (x) = λ x2 + μ x + 12, f' (4) = 15 and f' (2) = 11, then find λ and μ.
For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]
xn loga x
(x sin x + cos x ) (ex + x2 log x)
(1 − 2 tan x) (5 + 4 sin x)
x3 ex 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)
\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\]
\[\frac{a + \sin x}{1 + a \sin x}\]
\[\frac{x}{1 + \tan x}\]
\[\frac{x^5 - \cos x}{\sin x}\]
\[\frac{ax + b}{p x^2 + qx + r}\]
If |x| < 1 and y = 1 + x + x2 + x3 + ..., then write the value of \[\frac{dy}{dx}\]
Mark the correct alternative in of the following:
If \[f\left( x \right) = \frac{x - 4}{2\sqrt{x}}\]
(ax2 + cot x)(p + q cos x)
