Advertisements
Advertisements
Question
Differentiate of the following from first principle:
sin (x + 1)
Advertisements
Solution
\[ \frac{d}{dx}\left( f\left( x \right) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[\frac{d}{dx}\left( \sin \left( x + 1 \right) \right) = \lim_{h \to 0} \frac{\sin \left( x + h + 1 \right) - \sin \left( x + 1 \right)}{h}\]
\[\text{ We know }:\]
\[\sin C - \sin D = 2 \cos \left( \frac{C + D}{2} \right) \sin \left( \frac{C - D}{2} \right)\]
\[ = \lim_{h \to 0} \frac{2 \cos \left( \frac{x + h + 1 + x + 1}{2} \right) \sin \left( \frac{x + h + 1 - x - 1}{2} \right)}{h}\]
\[ = \lim_{h \to 0} \frac{2 \cos \left( \frac{2x + h + 2}{2} \right) \sin \left( \frac{h}{2} \right)}{h}\]
\[ = 2 \lim_{h \to 0} \cos \left( \frac{2x + h + 2}{2} \right) \lim_{h \to 0} \frac{\sin \left( \frac{h}{2} \right)}{\frac{h}{2}} \times \frac{1}{2}\]
\[ = 2 \cos \left( x + 1 \right) \times \frac{1}{2}\]
\[ = \cos \left( x + 1 \right)\]
APPEARS IN
RELATED QUESTIONS
Find the derivative of 99x at x = 100.
Find the derivative of x–3 (5 + 3x).
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):
`cos x/(1 + sin x)`
Find the derivative of f (x) = 3x at x = 2
Find the derivative of f (x) x at x = 1
Find the derivative of the following function at the indicated point:
2 cos x at x =\[\frac{\pi}{2}\]
\[\frac{x^2 - 1}{x}\]
\[\frac{x + 1}{x + 2}\]
k xn
\[\sqrt{2 x^2 + 1}\]
\[\frac{2x + 3}{x - 2}\]
Differentiate of the following from first principle:
\[\cos\left( x - \frac{\pi}{8} \right)\]
Differentiate of the following from first principle:
x cos x
Differentiate each of the following from first principle:
x2 sin x
Differentiate each of the following from first principle:
\[e^\sqrt{2x}\]
\[\frac{x^3}{3} - 2\sqrt{x} + \frac{5}{x^2}\]
(2x2 + 1) (3x + 2)
\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]
If for f (x) = λ x2 + μ x + 12, f' (4) = 15 and f' (2) = 11, then find λ and μ.
(x sin x + cos x ) (ex + x2 log x)
logx2 x
\[e^x \log \sqrt{x} \tan x\]
(2x2 − 3) sin x
\[\frac{x}{1 + \tan x}\]
\[\frac{1}{a x^2 + bx + c}\]
\[\frac{x^2 - x + 1}{x^2 + x + 1}\]
\[\frac{a + \sin x}{1 + a \sin x}\]
\[\frac{x}{1 + \tan x}\]
\[\frac{\sec x - 1}{\sec x + 1}\]
\[\frac{1}{a x^2 + bx + c}\]
If \[\frac{\pi}{2}\] then find \[\frac{d}{dx}\left( \sqrt{\frac{1 + \cos 2x}{2}} \right)\]
Write the value of \[\frac{d}{dx} \left\{ \left( x + \left| x \right| \right) \left| x \right| \right\}\]
Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.
Mark the correct alternative in of the following:
If \[f\left( x \right) = \frac{x - 4}{2\sqrt{x}}\]
Find the derivative of x2 cosx.
Find the derivative of f(x) = tan(ax + b), by first principle.
(ax2 + cot x)(p + q cos x)
