Advertisements
Advertisements
प्रश्न
Find the derivative of the following w. r. t. x by using method of first principle:
`x sqrt(x)`
उत्तर
Let f(x) = `x sqrt(x) = x^(3/2)`.
Then f(x + h) = `(x + "h")^(3/2)`
By definition,
f'(x) = `lim_("h" -> 0) ("f"(x + "h") - "f"(x))/"h"`
= `lim_("h" -> 0) ((x + "h")^(3/2) - x^(3/2))/"h"`
= `lim_("h" -> 0) ((x + "h")^(3/2) - x^(3/2))/"h" * ((x + "h")^(3/2) + x^3/2)/((x + "h")^(3/2) + x^(3/2))`
= `lim_("h" -> 0) ((x + "h")^3 - x^3)/("h"[(x + "h")^(3/2) + x^(3/2)]`
= `lim_(x -> 0) (x^3 + 3x^2"h" + 3x"h"^2 + "h"^3 - x^3)/("h"[(x + "h")^(3/2) + x^(3/2)]`
= `lim_("h" -> 0) ("h"(3x^2 + 3x"h" + "h"^2))/("h"[(x + "h")^(3/2) + x^(3/2)]`
= `lim_("h" -> 0) (3x^2 + 3x"h" + "h"^2)/((x + "h")^(3/2) + x^(3/2))` ...[h → 0, h ≠ 0]
= `(lim_("h" -> 0) (3x^2 + 3x"h" + "h"^2))/(lim_("h" -> 0) (x + "h")^(3/2) + lim_("h" -> 0) x^(3/2))`
= `(3x^2 + 3x xx 0 + 0^2)/((x + 0)^(3/2) + x^(3/2))`
= `(3x^2)/(2x^(3/2)`
= `3/2 sqrt(x)`
Alternative Method:
Let f(x) = `xsqrt(x) = x^(3/2)`.
Then f(x + h) = `(x + "h")^(3/2)`
∴ f'(x) = `lim_("h" -> 0) ("f"(x + "h") - "f"(x))/"h"`
= `lim_("h" -> 0) ((x + "h")^(3/2) - x^(3/2))/"h"`
Put x + h = y
∴ h = y – x and as h → 0, y → x
∴ f'(x) = `lim_(y -> x) (y^(3/2) - x^(3/2))/(y - x)`
= `3/2 x^(3/2 - 1) ...[because lim_(x -> "a") (x^"n" - "a"^"n")/(x - "a") = "na"^("n" - 1)]`
= `3/2 sqrt(x)`
APPEARS IN
संबंधित प्रश्न
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. : `x sqrtx + logx − e^x`
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:
sin (3x)
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:
sec (5x − 2)
Find the derivative of the following w. r. t. x. at the point indicated against them by using method of first principle:
tan x at x = `pi/4`
Find the derivative of the following w. r. t. x. at the point indicated against them by using method of first principle:
log(2x + 1) 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:
`"e"^(3x - 4)` 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) at x = 2
f(x) = [x] if x ∈ [0, 4). [where [*] is a greatest integer (floor) function]
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 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 whether the following function is differentiable at x = 3 where,
f(x) `{:(= x^2 + 2"," , "for" x ≥ 3),(= 6x - 7"," , "for" x < 3):}`
Find the values of p and q that make function f(x) differentiable everywhere on R
f(x) `{:( = 3 - x"," , "for" x < 1),(= "p"x^2 + "q"x",", "for" x ≥ 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):}`
Determine all real values of p and q that ensure the function
f(x) `{:( = "p"x + "q"",", "for" x ≤ 1),(= tan ((pix)/4)",", "for" 1 < x < 2):}` is differentiable at x = 1
Test whether the function f(x) `{:(= 2x - 3",", "for" x ≥ 2),(= x - 1",", "for" x < 2):}` is differentiable at x = 2
Test whether the function f(x) `{:(= x^2 + 1",", "for" x ≥ 2),(= 2x + 1",", "for" x < 2):}` is differentiable at x = 2
Test whether the function f(x) `{:(= 5x - 3x^2",", "for" x ≥ 1),(= 3 - x",", "for" x < 1):}` is differentiable at x = 1
