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:
tan x at x = `pi/4`
उत्तर
Let f(x) = tan x
∴ `"f"(pi/4) = tan pi/4`
`"f"(pi/4 + "h") = tan (pi/4 + "h")`
By definition,
`"f'"(pi/4) = lim_("h" -> 0) ("f"(pi/4 + "h") - "f"(pi/4))/"h"`
= `lim_("h" -> 0) (tan(pi/4 + "h") - tan pi/4)/"h"`
= `lim_("h" -> 0) (tan[(pi/4 + "h") - pi/4][1 + tan(pi/4 + "h") tan pi/4])/"h" ...[because tan("A" - "B") = (tan "A" - tan "B")/(1 + tan "A" tan "B")]`
= `lim_("h" -> 0) (tan"h"[1 + tan(pi/4 + "h") tan pi/4])/"h"`
= `[lim_("h" -> 0) tan"h"/"h"] xx [lim_("h" -> 0) {1 + tan(pi/4 + "h") tan pi/4}]`
= `1 xx [1 + tan(pi/4 + 0) tan pi/4]`
= 2 ...`[because tan pi/4 = 1]`
APPEARS IN
संबंधित प्रश्न
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. : x3 log x
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:
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:
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:
`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:
`"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) = x |x| at x = 0
Discuss the continuity and differentiability of f(x) = (2x + 3) |2x + 3| at x = `- 3/2`
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
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
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
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) `{:(= 5x - 3x^2",", "for" x ≥ 1),(= 3 - x",", "for" x < 1):}` is differentiable at x = 1
