Advertisements
Advertisements
Question
Differentiate of the following from first principle:
sin (2x − 3)
Advertisements
Solution
\[\frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\sin \left( 2x + 2h - 3 \right) - \sin \left( 2x - 3 \right)}{h}\]
\[\text{ We know }:\]
\[sin C-sin D=2 cos\left( \frac{C + D}{2} \right)\sin\left( \frac{C - D}{2} \right)\]
\[ = \lim_{h \to 0} \frac{2 \cos \left( \frac{2x + 2h - 3 + 2x - 3}{2} \right) \sin \left( \frac{2x + 2h - 3 + 2x - 3}{2} \right)}{h}\]
\[ = \lim_{h \to 0} \frac{2 \cos \left( \frac{4x + 2h - 6}{2} \right) \sin \left( h \right)}{h}\]
\[ = \lim_{h \to 0} 2 \cos \left( \frac{4x + 2h - 6}{2} \right) \lim_{h \to 0} \frac{\sin h}{h}\]
\[ = 2 \cos \left( \frac{4x - 6}{2} \right) \left( 1 \right)\]
\[ = 2 \cos \left( 2x - 3 \right)\]
\[ \]
\[\]
APPEARS IN
RELATED QUESTIONS
Find the derivative of x at x = 1.
Find the derivative of x–4 (3 – 4x–5).
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):
sin (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):
`(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 f (x) = 3x at x = 2
\[\frac{x^2 + 1}{x}\]
\[\frac{x^2 - 1}{x}\]
\[\frac{x + 2}{3x + 5}\]
Differentiate each of the following from first principle:
e−x
Differentiate each of the following from first principle:
\[\sqrt{\sin (3x + 1)}\]
Differentiate each of the following from first principle:
\[e^\sqrt{2x}\]
Differentiate each of the following from first principle:
\[e^\sqrt{ax + b}\]
\[\tan \sqrt{x}\]
log3 x + 3 loge x + 2 tan x
a0 xn + a1 xn−1 + a2 xn−2 + ... + an−1 x + an.
\[\frac{1}{\sin x} + 2^{x + 3} + \frac{4}{\log_x 3}\]
\[\frac{(x + 5)(2 x^2 - 1)}{x}\]
x2 ex log x
xn tan x
xn loga x
\[\frac{2^x \cot x}{\sqrt{x}}\]
x5 ex + x6 log x
(1 +x2) cos x
x5 (3 − 6x−9)
x−4 (3 − 4x−5)
\[\frac{e^x - \tan x}{\cot x - x^n}\]
\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\]
\[\frac{2^x \cot x}{\sqrt{x}}\]
\[\frac{1 + \log x}{1 - \log x}\]
If x < 2, then write the value of \[\frac{d}{dx}(\sqrt{x^2 - 4x + 4)}\]
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:
Let f(x) = x − [x], x ∈ R, then \[f'\left( \frac{1}{2} \right)\]
Mark the correct alternative in each of the following:
If\[f\left( x \right) = \frac{x^n - a^n}{x - a}\] then \[f'\left( a \right)\]
Mark the correct alternative in of the following:
If f(x) = x sinx, then \[f'\left( \frac{\pi}{2} \right) =\]
