Advertisements
Advertisements
प्रश्न
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
उत्तर
Let, f(x) = `"e"^(3x - 4)`
∴ f(2) = `"e"^(3(2) - 4)` = e2 and
f(2 + h) = `"e"^(3(2 + "h")-4` = e3h+2
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) ("e"^(3"h" + 2) - "e"^2)/"h"`
= `lim_("h" -> 0) ("e"^(3"h") "e"^2 - "e"^2)/"h"`
= `lim_("h" -> 0) ("e"^2 ("e"^(3"h") - 1))/"h"`
= `"e"^2 lim_("h" -> 0) (("e"^(3"h") - 1)/(3"h")) xx 3`
= `3"e"^2 (1) ...[because lim_(x -> 0) ("e"^("p"x) - 1)/("p"x) = 1]`
= 3e2
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)`
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. : `2/7 x^(7/2) + 5/2 x^(2/5)`
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:
e2x+1
Find the derivative of the following w. r. t. x by using method of first principle:
3x
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:
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`
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) 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) `{:(= 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^50/50 + x^49/49 + x^48/48 + .... +x^2/2 + x + 1`, thef f'(1) =
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):}`
Discuss whether the function f(x) = |x + 1| + |x – 1| is differentiable ∀ x ∈ R
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
