Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board chapter 9 - Differentiation [Latest edition]

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:

e^2x+1

Find the derivative of the following w. r. t. x by using method of first principle:

3^x 

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:

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:

`2^(3x + 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:

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

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) = (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

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`

Examine the function

f(x) `{:(= x^2 cos (1/x)",", "for"  x ≠ 0),(= 0",", "for"  x = 0):}`

for continuity and differentiability at x = 0

Differentiate the following w.r.t.x :

y = `x^(4/3) + "e"^x - sinx`

Differentiate the following w.r.t.x :

y = `sqrt(x) + tan x - x^3`

Differentiate the following w.r.t.x :

y = `log x - "cosec"  x + 5^x - 3/(x^(3/2))`

Differentiate the following w.r.t.x :

y = `x^(7/3) + 5x^(4/5) - 5/(x^(2/5))`

Differentiate the following w.r.t.x :

y = `7^x + x^7 - 2/3 xsqrt(x) - logx + 7^7`

Differentiate the following w.r.t.x :

y = `3 cotx - 5"e"^x + 3logx - 4/(x^(3/4))`

Differentiate the following w.r.t.x. :

y = x⁵ tan x

Differentiate the following w.r.t.x. :

y = x³ log x

Differentiate the following w.r.t.x. :

y = (x² + 2)² sin x

Differentiate the following w.r.t.x. :

y = e^x logx

Differentiate the following w.r.t.x. :

y = `x^(3/2)  "e"^xlogx`

Differentiate the following w.r.t.x. :

y = log e^_x3 log x³ 

Differentiate the following w.r.t.x. :

y = `x^2sqrt(x) + x^4logx`

Differentiate the following w.r.t.x. :

y = `"e"^xsecx - x^(5/3) log x`

Differentiate the following w.r.t.x. :

y = `x^4 + x sqrt(x) cos x - x^2"e"^x`

Differentiate the following w.r.t.x. :

y = (x³ – 2) tan x – x cos x + 7^x. x⁷

Differentiate the following w.r.t.x. :

y = `sinx logx + "e"^x cos x - "e"^x sqrt(x)`

Differentiate the following w.r.t.x. :

y = `"e"^x tanx + cos x log x - sqrt(x)  5^x`

Differentiate the following w.r.t.x. :

y = `(x^2 + 3)/(x^2 - 5)`

Differentiate the following w.r.t.x. :

y = `(sqrt(x) + 5)/(sqrt(x) - 5)`

Differentiate the following w.r.t.x. :

y = `(x"e"^x)/(x + "e"^x)`

Differentiate the following w.r.t.x. :

y = `(x log x)/(x + log x)`

Differentiate the following w.r.t.x. :

y = `(x^2 sin x)/(x + cos x)`

Differentiate the following w.r.t.x. :

y = `(5"e"^x - 4)/(3"e"^x - 2)`

If f(x) is a quadratic polynomial such that f(0) = 3, f'(2) = 2 and f'(3) = 12 then find f(x)

If f(x) = a sin x – b cos x, `"f'"(pi/4) = sqrt(2) and "f'"(pi/6)` = 2, then find f(x)

Fill in the blanks:

y = e^x .tan x

Differentiating w.r.t.x

`("d"y)/("d"x) = "d"/("d"x)("e"^x tan x)`

= `square "d"/("d"x) tanx + tan x "d"/("d"x) square`

= `square  square + tan x  square`

= `"e"^x [square  + square]`

Fill in the blanks:

y = `sinx/(x^2 + 2)`

Differentiating. w.r.t.x.

`("d"y)/("d"x) = (square "d"/("d"x) (sin x) - sin x "d"/("dx) square)/(x^2 + 2)^2`

= `(square  square - sin x  square)/(x^2 + 2)^2`

= `(square - square)/(x^2 + 2)^2`

Fill in the blanks:

y = (3x² + 5) cos x

Differentiating w.r.t.x

`("d"y)/("d"x) = "d"/("d"x) [(3x^2 + 5) cos x]`

= `(3x^2 + 5) "d"/("d"x) [square] + cos x  "d"/("d"x) [square]`

= `(3x^2 + 5) [square] + cos x  [square]`

∴ `(dx)/("d"y) = (3x^2 + 5) [square] + [square] cos x`

Fill in the blank:

Differentiate tan x and sec x w.r.t.x. using the formulae for differentiation of `"u"/"v" and 1/"v"` respectively

Select the correct answer from the given alternative:

If y = `(x - 4)/(sqrtx + 2)`, then `("d"y)/("d"x)`

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 y = `(3x + 5)/(4x + 5)`, then `("d"y)/("d"x)` =

Select the correct answer from the given alternative:

If y = `(5sin x - 2)/(4sin x + 3)`, then `("d"y)/("d"x)` =

Select the correct answer from the given alternative:

Suppose f(x) is the derivative of g(x) and g(x) is the derivative of h(x).

If h(x) = a sin x + b cos x + c then f(x) + h(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 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

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 f(2) = 4, f′(2) = 1 then find `lim_(x -> 2) [(x"f"(2) - 2"f"(x))/(x - 2)]`

If y = `"e"^x/sqrt(x)` find `("d"y)/("d"x)` when x = 1