Advertisements
Advertisements
प्रश्न
Find the derivative of the following function:
2tan x – 7sec x
Advertisements
उत्तर
Let f (x) = 2 tan x – 7 sec x Accordingly, from the first principle,
f'(x) = `lim_(h->0) (f(x + h) - f(x))/h`
= `lim_(h->0)1/h [2tan (x + h) - 7 sec (x + h) -2 tan x + 7sec x]`
= `2 lim_(h->0)1/h[2{tan (x + h) - tan x} - 7{sec (x + h) -sec x}]`
= `2 lim_(h->0)1/h[tan (x + h) - tan x] - 7lim_(h->0)1/h[sec (x + h) -sec x]`
= `2 lim_(h->0) 1/h [sin (x + h)/(cos (x + h))-(sinx)/(cos x)] -7 lim_(h->0)1/h [1/(cos (x + h)) - 1/(cos x)]`
= `2 lim_(h->0)1/h [(sin(x + h) cos x - sin x cos (x + h))/(cos x cos (x + h))] -7lim_(h->0)1/h[(cos x - cos (x + h))/(cos x cos (x + h))]`
= `2 lim_(h->0) [(sin (x + h - x))/(cos x cos (x + h))] -7 lim_(h->0)[(-2 sin ((x + x + h)/2) sin ((x - x - h)/(2)))/(cos x cos (x + h))]`
= `2 lim_(h->0) [((sin h)/h) 1/(cos x cos (x +h))] - 7 lim_(h->0)1/h[(-2 sin ((2x + h)/h) sin (-h/2))/(cos x cos (x + h))]`
= `2 (lim_(h->0)(sin h)/h) (lim_(h->0) 1/(cos x cos (x + h)))-7 (lim_(h->0) (sin h/2)/(h/2)) (lim_(h ->0) (sin ((2x + h)/2))/(cos x cos (x + h)))`
= 2.1 `1/(cos x cos x) - 7.1 (sin x/(cos x cos x))`
= 2 sec2 x - 7 sec x tan x
APPEARS IN
संबंधित प्रश्न
Find the derivative of `x^n + ax^(n-1) + a^2 x^(n-2) + ...+ a^(n -1) x + a^n` for some fixed real number a.
For some constants a and b, find the derivative of (x – a) (x – b).
For some constants a and b, find the derivative of `(x - a)/(x - b)`.
Find the derivative of cos x from first principle.
Find the derivative of the following function:
cosec x
Find the derivative of the following function:
5sin x – 6cos x + 7
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):
(ax2 + sin x) (p + q cos 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):
`(x + cos x)(x - tan 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):
`(x^2 cos (pi/4))/sin 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):
`x/(sin^n x)`
Find the derivative of f(x) = ax + b, where a and b are non-zero constants, by first principle
Find the derivative of f(x) = ax2 + bx + c, where a, b and c are none-zero constant, by first principle.
Find the derivative of f(x) = sin x, by first principle.
`(x^4 + x^3 + x^2 + 1)/x`
`(x + 1/x)^3`
`(3x + 4)/(5x^2 - 7x + 9)`
`(x^5 - cosx)/sinx`
(sin x + cos x)2
x2 sin x + cos 2x
sin3x cos3x
If `y = sqrt(x) + 1/sqrt(x)`, then`(dy)/(dx)` at x = 1 is ______.
if `f(x) = (x - 4)/(2sqrt(x))`, then f'(1) is ______.
If `y = (1 + 1/x^2)/(1 - 1/x^2)` then `(dy)/(dx)` is ______.
If `y = (sin x + cos x)/(sin x - cos x)`, then `(dy)/(dx)` at x = 0 is ______.
If `y = (sin(x + 9))/cosx` then `(dy)/(dx)` at x = 0 is ______.
If `f(x) = 1 + x + x^2/2 + ... + x^100/100`, then f'(1) is equal to ______.
If `f(x) = x^100 + x^99 .... + x + 1`, then f'(1) is equal to ______.
If `f(x) = 1 - x + x^2 - x^3 + ... -x^99 + x^100`, then f'(1) is equal to ______.
If `y = 1 + x/(1!) + x^2/(2!) + x^3/(3!) + ...,` then `(dy)/(dx)` = ______.
