Advertisements
Advertisements
प्रश्न
Find the derivative of f(x) = tan(ax + b), by first principle.
Advertisements
उत्तर
We have f'(x) = `lim_(h -> 0) (f(x + h) - f(x))/h`
= `lim_(h -> 0) (tan(a(x + h) + b) - tan(ax + b))/h`
= `lim_(h -> 0) ((sin(ax + ah + b))/(cos(ax + ah + b)) - (sin(ax + b))/(cos(ax + b)))/h`
= `lim_(h -> 0) (sin(ax + ah + b) cos(ax + b) - sin(ax + b) cos(ax + ah + b))/(h cos(ax + b) cos(ax + ah + b))`
= `lim_(h -> 0) (a sin (ah))/(a * h cos (ax + b) cos(ax + ah + b))`
= `lim_(h -> 0) a/(cos(ax + b) cos(ax + ah + b))`
= `lim_(ah -> 0) (sin ah)/(ah)` ....[as h → 0 ah → 0]
= `a/(cos^2 (ax + b))`
= `a sec^2 (ax + b)`.
APPEARS IN
संबंधित प्रश्न
Find the derivative of 99x at x = 100.
Find the derivative of x–3 (5 + 3x).
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/(ax^2 + bx + c)`
Find the derivative of f (x) = cos x at x = 0
Find the derivative of the following function at the indicated point:
Find the derivative of the following function at the indicated point:
sin 2x at x =\[\frac{\pi}{2}\]
x ex
Differentiate of the following from first principle:
(−x)−1
Differentiate of the following from first principle:
sin (2x − 3)
Differentiate each of the following from first principle:
\[\sqrt{\sin (3x + 1)}\]
Differentiate each of the following from first principle:
sin x + cos x
tan (2x + 1)
\[\frac{x^3}{3} - 2\sqrt{x} + \frac{5}{x^2}\]
(2x2 + 1) (3x + 2)
\[\frac{2 x^2 + 3x + 4}{x}\]
a0 xn + a1 xn−1 + a2 xn−2 + ... + an−1 x + an.
\[\frac{1}{\sin x} + 2^{x + 3} + \frac{4}{\log_x 3}\]
cos (x + a)
Find the rate at which the function f (x) = x4 − 2x3 + 3x2 + x + 5 changes with respect to x.
For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]
\[e^x \log \sqrt{x} \tan x\]
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.
\[\frac{2x - 1}{x^2 + 1}\]
\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\]
\[\frac{e^x + \sin x}{1 + \log x}\]
\[\frac{\sin x - x \cos x}{x \sin x + \cos x}\]
\[\frac{a + \sin x}{1 + a \sin x}\]
\[\frac{4x + 5 \sin x}{3x + 7 \cos x}\]
Write the value of \[\frac{d}{dx}\left( x \left| x \right| \right)\]
If f (x) = |x| + |x−1|, write the value of \[\frac{d}{dx}\left( f (x) \right)\]
Write the value of \[\frac{d}{dx}\left( \log \left| x \right| \right)\]
If f (x) = \[\log_{x_2}\]write the value of f' (x).
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 of the following:
If \[f\left( x \right) = \frac{x - 4}{2\sqrt{x}}\]
Let f(x) = x – [x]; ∈ R, then f'`(1/2)` is ______.
