Advertisements
Advertisements
Question
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
Advertisements
Solution
Let f(x) = `sqrt(2x + 5)`
∴ f(2) = `sqrt(2(2) + 5) = sqrt(9)` = 3 and
f(2 + h) = `sqrt(2(2 + "h") + 5) = sqrt(2"h" + 9)`
By first principle, we get
f'(a) = `lim_("h" -> 0) ("f"("a" + "h") - "f"("a"))/"h"`
∴ f'(2) = `lim_("h" -> 0) ("f"(2 + "h") - "f"(2))/"h"`
= `lim_("h" -> 0) (sqrt(2"h" + 9) - 3)/"h"`
= `lim_("h" -> 0) (sqrt(2"h" + 9) - 3)/"h" xx (sqrt(2"h" + 9) + 3)/(sqrt(2"h" + 9) + 3)`
= `lim_("h" -> 0) (2"h" + 9 - 9)/("h"(sqrt(2"h" + 9) + 3)`
= `lim_("h" -> 0) (2"h")/("h"(sqrt(2"h" + 9) + 3)`
= `lim_("h" -> 0) 2/(sqrt(2"h" + 9) + 3)` ...[∵ h → 0, ∴ h ≠ 0]
= `2/(sqrt(0 + 9) + 3)`
= `2/(3 + 3)`
= `1/3`
APPEARS IN
RELATED QUESTIONS
Find the derivative of the following function w. r. t. x.:
`7xsqrt x`
Find the derivative of the following function w. r. t. x.:
35
Differentiate the following w. r. t. x.: x5 + 3x4
Differentiate the following w. r. t. x. : `x^(5/2) + 5x^(7/5)`
Differentiate the following w. r. t. x. : `x^(5/2) e^x`
Differentiate the following w. r. t. x. : ex log 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:
sin (3x)
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:
`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:
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:
cos x at x = `(5pi)/4`
Discuss the continuity and differentiability of f(x) = x |x| 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]
Test the continuity and differentiability of f(x) `{:(= 3 x + 2, "if" x > 2),(= 12 - x^2, "if" x ≤ 2):}}` at x = 2
If f(x) `{:(= sin x - cos x, "if" x ≤ pi/2),(= 2x - pi + 1, "if" x > pi /2):}` Test the continuity and differentiability of f at x = `π/2`
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) `{:( = x^2 + sin x + 1, "for" x ≤ 0),(= x^2 - 2x + 1, "for" x ≤ 0):}` then
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 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
Discuss whether the function f(x) = |x + 1| + |x – 1| is differentiable ∀ x ∈ R
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
If y = `"e"^x/sqrt(x)` find `("d"y)/("d"x)` when x = 1
