Advertisements
Advertisements
प्रश्न
\[\tan \sqrt{x}\]
Advertisements
उत्तर
\[text{ Let } f(x) = \tan x^2 \]
\[\text{ Thus, we have }: \]
\[f(x + h) = \tan (x + h )^2 \]
\[\frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\]
\[ = \lim_{h \to 0} \frac{\tan (x + h )^2 - \tan x^2}{h}\]
\[ = \lim_{h \to 0} \frac{\sin\left( \left( x + h \right)^2 - x^2 \right)}{h \cos \left( x + h \right)^2 \cos x^2} \left[ \because \tan A - \tan B = \frac{\sin (A - B)}{\cos A \cos B} \right]\]
\[ = \lim_{h \to 0} \frac{\sin( x^2 + h^2 + 2hx - x^2 )}{h\cos \left( x + h \right)^2 \cos x^2}\]
\[ = \lim_{h \to 0} \frac{\sin(h\left( h + 2x) \right)}{h\left( h + 2x \right) \cos \left( x + h \right)^2 \cos x^2} \times \left( h + 2x \right)\]
\[ = \lim_{h \to 0} \frac{\sin(h\left( h + 2x) \right)}{(h\left( h + 2x) \right)} \lim_{h \to 0} \frac{h + 2x}{\cos(x + h )^2 \cos x^2} \left[ As \lim_{h \to 0} \frac{\sin(h\left( h + 2x) \right)}{(h\left( h + 2x) \right)} = 1 \right]\]
\[ = 1 \times \frac{2x}{\cos^2 x^2}\]
\[ = 2x \sec^2 x^2 \]
\[\]
APPEARS IN
संबंधित प्रश्न
Find the derivative of (5x3 + 3x – 1) (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):
`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):
(ax + b)n
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):
cosec x cot 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):
`(a + bsin x)/(c + dcosx)`
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 + a))/ cos x`
Find the derivative of the following function at the indicated point:
sin x at x =\[\frac{\pi}{2}\]
\[\frac{1}{\sqrt{3 - x}}\]
(x2 + 1) (x − 5)
(x2 + 1) (x − 5)
Differentiate each of the following from first principle:
e−x
Differentiate of the following from first principle:
sin (x + 1)
Differentiate each of the following from first principle:
\[\frac{\cos x}{x}\]
tan (2x + 1)
\[\sin \sqrt{2x}\]
\[\cos \sqrt{x}\]
\[\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^3\]
a0 xn + a1 xn−1 + a2 xn−2 + ... + an−1 x + an.
\[\frac{1}{\sin x} + 2^{x + 3} + \frac{4}{\log_x 3}\]
\[\text{ If } y = \left( \frac{2 - 3 \cos x}{\sin x} \right), \text{ find } \frac{dy}{dx} at x = \frac{\pi}{4}\]
\[If y = \sqrt{\frac{x}{a}} + \sqrt{\frac{a}{x}}, \text{ prove that } 2xy\frac{dy}{dx} = \left( \frac{x}{a} - \frac{a}{x} \right)\]
If for f (x) = λ x2 + μ x + 12, f' (4) = 15 and f' (2) = 11, then find λ and μ.
sin x cos x
sin2 x
logx2 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) (a + d)2
\[\frac{x^2 + 1}{x + 1}\]
\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\]
\[\frac{1 + 3^x}{1 - 3^x}\]
\[\frac{\sec x - 1}{\sec x + 1}\]
\[\frac{x + \cos x}{\tan x}\]
\[\frac{x}{\sin^n x}\]
\[\frac{ax + b}{p x^2 + qx + r}\]
Write the value of \[\lim_{x \to a} \frac{x f (a) - a f (x)}{x - a}\]
Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.
If f (x) = |x| + |x−1|, write the value of \[\frac{d}{dx}\left( f (x) \right)\]
