Advertisements
Advertisements
Question
Find the derivative of the following w. r. t. x by using method of first principle:
sin (3x)
Solution
Let f(x) = sin 3x
∴ f(x + h) = sin [3(x + h)] = sin (3x + 3h)
f(x + h) – f(x) = sin(3x + 3h) – sin 3x
= `2cos((3x + 3"h" + 3x)/2)*sin((3x + 3"h" - 3x)/2)`
= `2cos((6x + 3"h")/2)*sin ((3"h")/2)`
By definition,
f'(x) = `lim_("h"-> 0) ("f"(x + "h") - "f"(x))/"h"`
= `lim_("h" -> 0) (2cos((6x + 3"h")/2)*sin((3"h")/2))/ "h"`
= `lim_("h" -> 0) 2[cos ((6x + 3"h")/2)]*[(sin ((3"h")/2))/(((3"h")/2))] xx 3/2`
= `3[lim_("h" -> 0) cos((6x + 3"h")/2)] xx [lim_("h" -> 0) (sin((3"h")/2))/(((3"h")/2))]`
= `3[cos ((6x + 0)/2) ] xx 1 ...[because "h" -> 0"," (3"h")/2 -> 0 "and" lim_(theta-> 0) sintheta/theta = 1]`
3 cos 3x
APPEARS IN
RELATED QUESTIONS
Find the derivative of the following function w.r.t. x.:
x–9
Find the derivative of the following functions w. r. t. x.:
`x^(3/2)`
Differentiate the following w. r. t. x.: x5 + 3x4
Differentiate the following w. r. t. x. : `x sqrtx + logx − e^x`
Differentiate the following w. r. t. x. : `x^(5/2) + 5x^(7/5)`
Differentiate the following w. r. t. x. : `2/7 x^(7/2) + 5/2 x^(2/5)`
Differentiate the following w. r. t. x. : x3 log x
Differentiate the following w. r. t. x. : `x^(5/2) e^x`
Differentiate the following w. r. t. x. : x3 .3x
Find the derivative of the following w. r. t. x by using method of first principle:
x2 + 3x – 1
Find the derivative of the following w. r. t. x by using method of first principle:
e2x+1
Find the derivative of the following w. r. t. x by using method of first principle:
log (2x + 5)
Find the derivative of the following w. r. t. x by using method of first principle:
tan (2x + 3)
Find the derivative of the following w. r. t. x by using method of first principle:
sec (5x − 2)
Find the derivative of the following w. r. t. x by using method of first principle:
`x sqrt(x)`
Find the derivative of the following w. r. t. x. at the point indicated against them by using method of first principle:
`sqrt(2x + 5)` at x = 2
Find the derivative of the following w. r. t. x. at the point indicated against them by using method of first principle:
`2^(3x + 1)` at x = 2
Show that the function f is not differentiable at x = −3, where f(x) `{:(= x^2 + 2, "for" x < - 3),(= 2 - 3x, "for" x ≥ - 3):}`
Show that f(x) = x2 is continuous and differentiable at x = 0
Discuss the continuity and differentiability of f(x) = x |x| at x = 0
Discuss the continuity and differentiability of f(x) = (2x + 3) |2x + 3| at x = `- 3/2`
Test the continuity and differentiability of f(x) `{:(= 3 x + 2, "if" x > 2),(= 12 - x^2, "if" x ≤ 2):}}` at x = 2
Examine the function
f(x) `{:(= x^2 cos (1/x)",", "for" x ≠ 0),(= 0",", "for" x = 0):}`
for continuity and differentiability at x = 0
Select the correct answer from the given alternative:
If y = `("a"x + "b")/("c"x + "d")`, then `("d"y)/("d"x)` =
Select the correct answer from the given alternative:
If f(x) `{:(= 2x + 6, "for" 0 ≤ x ≤ 2),(= "a"x^2 + "b"x, "for" 2 < x ≤4):}` is differentiable at x = 2 then the values of a and b are
Select the correct answer from the given alternative:
If f(x) `{:( = x^2 + sin x + 1, "for" x ≤ 0),(= x^2 - 2x + 1, "for" x ≤ 0):}` then
Select the correct answer from the given alternative:
If, f(x) = `x^50/50 + x^49/49 + x^48/48 + .... +x^2/2 + x + 1`, thef f'(1) =
Determine the values of p and q that make the function f(x) differentiable on R where
f(x) `{:( = "p"x^3",", "for" x < 2),(= x^2 + "q"",", "for" x ≥ 2):}`
Test whether the function f(x) `{:(= x^2 + 1",", "for" x ≥ 2),(= 2x + 1",", "for" x < 2):}` is differentiable at x = 2
