NCERT solutions for Class 12 Maths chapter 5 - Continuity and Differentiability [Latest edition]

Prove that the function f (x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5.

Examine the following functions for continuity.

`f(x) = (x^2 - 25)/(x + 5), x != -5`

Examine the continuity of the function f (x) = 2x² – 1 at x = 3.

Examine the following functions for continuity. 

f (x) = x – 5

Examine the following functions for continuity

`1/(x - 5), x != 5`

Examine the following functions for continuity

f(x) = | x – 5|

Prove that the function `f(x) = x^n` is continuous at x = n, where n is a positive integer

Is the function f defined by f(x)= `{(x, if x<=1),(5, if x > 1):}` 

continuous at x = 0? At x = 1? At x = 2?

Find all points of discontinuity of f, where f is defined by

`f(x) = {(2x +3, if zx <=2),(2x - 3, if x > 2):}`

Find all points of discontinuity of f, where f is defined by `f(x) = {(|x|+3, if x<= -3),(-2x, if -3 < x < 3),(6x + 2, if x >= 3):}`

Find all points of discontinuity of f, where f is defined by `f(x) = {(|x|/x , if x != 0),(0, if x = 0):}`

Find all points of discontinuity of f, where f is defined by

`f(x) = {(x/|x|, ","if x < 0),(-1, ","if x >= 0):}`

Find all points of discontinuity of f, where f is defined by 

`f(x) = {(x+1, "," if x >= 1),(x^2 + 1, ","if x < 1):}`

Find all points of discontinuity of f, where f is defined by `f(x) = {(x^3 - 3, if x <= 2),(x^2 + 1, if x > 2):}`

Find all points of discontinuity of f, where f is defined by `f(x) = {(x^10 - 1, ","if x <= 1),(x^2, ","if x > 1):}`

Is the function defined by

`f(x) = {(x+5, if x <= 1),(x -5, if x > 1):}` a continuous function?

Discuss the continuity of the function f, where f is defined by

`f(x) = {(3, ","if 0 <= x <= 1),(4, ","if 1 < x < 3),(5, ","if 3 <= x <= 10):}`

Discuss the continuity of the function f, where f is defined by

`f(x) = {(2x , ","if x < 0),(0, "," if 0 <= x <= 1),(4x, "," if x > 1):}`

Discuss the continuity of the function f, where f is defined by

`f(x) = {(-2,"," if x <= -1),(2x, "," if -1 < x <= 1),(2, "," if x > 1):}`

Find the relationship between a and b so that the function f defined by `f(x)= {(az + 1, if x<= 3),(bx + 3, if x  > 3):}` is continuous at x = 3.

For what value of `lambda` is the function defined by

`f(x) = {(lambda(x^2 - 2x),  "," if x <= 0),(4x+ 1, "," if x > 0):}`  continuous at x = 0? What about continuity at x = 1?

Show that the function defined by  g(x) = x = [x] is discontinuous at all integral point. Here [x] denotes the greatest integer less than or equal to x.

Is the function defined by  `f(x) = x^2 - sin x + 5` continuous at x = π? 

Discuss the continuity of the following functions.

(a) f (x) = sin x + cos x

(b) f (x) = sin x − cos x

(c) f (x) = sin x × cos x

Find the points of discontinuity of f, where

`f(x) = {((sinx)/x, "," if x < 0),(x + 1, "," if x >= 0):}`

Determine if f defined by

`f(x) = {(x^2 sin  1/x, "," if x != 0),(0, "," if x = 0):}` is a continuous function?

Examine the continuity of f, where f is defined by

`f(x) = {(sin x - cos x, if x != 0),(-1, "," if x = 0):}`

Find the values of k so that the function f is continuous at the indicated point.

`f(x) = {((kcosx)/(pi-2x), "," if x != pi/2),(3, "," if x = pi/2):}  " at x ="  pi/2` 

Find the values of k so that the function f is continuous at the indicated point.

`f(x) = {(kx^2, "," if x<= 2),(3, "," if x > 2):} " at x" = 2`

Find the values of k so that the function f is continuous at the indicated point.

`f(x) = {(kx +1, if x<= pi),(cos x, if x > pi):} " at  x " = pi`

Find the values of k so that the function f is continuous at the indicated point.

`f(x) = {(kx + 1, "," if x <= 5),(3x - 5, "," if x > 5):} " at x " = 5`

Find the values of a and b such that the function defined by

`f(x) = {(5, "," if x <= 2),(ax +b, "," if 2 < x < 10),(21, "," if x >= 10):}` 

Differentiate the functions with respect to x.

sin (x² + 5)

Differentiate the functions with respect to x.

cos (sin x)

Differentiate the functions with respect to x.

sin (ax + b)

Differentiate the functions with respect to x.

`sec(tan (sqrtx))`

Differentiate the functions with respect to x.

`(sin (ax + b))/cos (cx + d)`

Differentiate the functions with respect to x. 

`cos x^3. sin^2 (x^3)`

Differentiate the functions with respect to x. 

`2sqrt(cot(x^2))`

Differentiate the functions with respect to x.

`cos (sqrtx)`

Prove that the function f given by  `f(x) = |x - 1|, x  in R`  is notdifferentiable at x = 1.

Find  `dy/dx`

2x + 3y = sin x

Find `dy/dx`

2x + 3y = sin y

Find `dy/dx`

ax + by² = cos y

Find `dy/dx`

xy + y² = tan x + y

Find `dx/dy`

x² + xy + y2 = 100

Find `dy/dx`

x³ + x2y + xy² + y³ = 81

Find `dy/dx`

sin² y + cos xy = Π

Find `dy/dx`

sin² x + cos² y = 1

Find `dy/dx`

`y = sin^(-1)((2x)/(1+x^2))`

Find `dy/dx`

`y = tan^(-1) ((3x -x^3)/(1 - 3x^2)), - 1/sqrt3 < x < 1/sqrt3`

Find `dy/dx`

`y = cos^(-1) ((1-x^2)/(1+x^2)), 0 < x < 1`

Find `dy/dx`

`y = sin^(-1) ((1-x^2)/(1+x^2)), 0 < x < 1`

Find `dx/dy`

`y = cos^(-1) ((2x)/(1+x^2)), -1 < x < 1`

Find `dy/dx`

`y = sin^(-1)(2xsqrt(1-x^2)), -1/sqrt2 < x  < 1/sqrt2`

Find `dy/dx`

`y = sec^(-1) (1/(2x^2 - 1)), 0 < x < 1/sqrt2`

Differentiate the following w.r.t. x:

`e^x/sinx`

Differentiate the following w.r.t. x:  `e^(sin^(-1) x)`

Differentiate the following w.r.t. x: `e^(x^3)`

Differentiate the following w.r.t. x: 

sin (tan–1 e^–x)

Differentiate the following w.r.t. x:

`log(cos e^x)`

Differentiate the following w.r.t. x:

`e^x + e^(x^2) + ....+ e^(x^3)`

Differentiate the following w.r.t. x:

`sqrt(e^(sqrtx)), x > 0`

Differentiate the following w.r.t. x: log (log x), x > 1

Differentiate the following w.r.t. x: 

`cos x/log x, x >0`

Differentiate the following w.r.t. x:

cos (log x + e^x), x > 0

Differentiate the function with respect to x. 

cos x . cos 2x . cos 3x

Differentiate the function with respect to x.

`sqrt(((x-1)(x-2))/((x-3)(x-4)(x-5)))`

Differentiate the function with respect to x.

`(log x)^(cos x)`

Differentiate the function with respect to x.

`x^x - 2^(sin x)`

Differentiate the function with respect to x.

(x + 3)² . (x + 4)³ . (x + 5)⁴

Differentiate the function with respect to x.

`(x + 1/x)^x + x^((1+1/x))`

Differentiate the function with respect to x.

(log x)^x + xl^{og x}

Differentiate the function with respect to x.

`(sin x)^x + sin^(-1) sqrtx`

Differentiate the function with respect to x.

x^{sin x} + (sin x)^{cos x}

Differentiate the function with respect to x.

`x^(xcosx) + (x^2 + 1)/(x^2 -1)`

Differentiate the function with respect to x.

`(x cos x)^x + (x sin x)^(1/x)`

Find `dy/dx` of function

x^y + y^x = 1

Find `dy/dx` of Function y^x = x^y

Find `dy/dx` of Function

(cos x)^y = (cos y)^x

Find `dy/dx` of function

xy = e^{(x – y)}

Find the derivative of the function given by f (x) = (1 + x) (1 + x²) (1 + x⁴) (1 + x⁸) and hence find f ′(1).

Differentiate (x² – 5x + 8) (x³ + 7x + 9) in three ways mentioned

(i) by using product rule

(ii) by expanding the product to obtain a single polynomial.

(iii) by logarithmic differentiation.

Do they all give the same answer?d below:

If u, v and w are functions of x, then show that

`d/dx(u.v.w) = (du)/dx v.w+u. (dv)/dx.w + u.v. (dw)/dx`

in two ways-first by repeated application of product rule, second by logarithmic differentiation.

If x and y are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

`x = 2at^2, y = at^4`

If x and y are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

x = sin t, y = cos 2t

If x and y are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

x = 4t, y = 4/y

If x and y are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

x = cos θ – cos 2θ, y = sin θ – sin 2θ

If x and y are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

x = a cos θ, y = b cos θ

If x and y are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

x = a (θ – sin θ), y = a (1 + cos θ)

If x and y are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

`x = (sin^3t)/sqrt(cos 2t),  y  = (cos^3t)/sqrt(cos 2t)`

If x and y are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

`x = a(cos t + log tan  t/2), y =  a sin t`

If x and y are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

x = a sec θ, y = b tan θ

If x and y are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

x = a (cos θ + θ sin θ), y = a (sin θ – θ cos θ)

if `x = sqrt(a^(sin^(-1)))`, y = `sqrt(a^(cos^(-1)))` show that `dy/dx = - y/x`

Find the second order derivatives of the function.

x² + 3x + 2

Find the second order derivatives of the function.

x . cos x

Find the second order derivatives of the function.

log x

Find the second order derivatives of the function.

x³ log x

Find the second order derivatives of the function.

e^x sin 5x

Find the second order derivatives of the function.

e^6x cos 3x

Find the second order derivatives of the function.

tan^–1 x

Find the second order derivatives of the function.

log (log x)

Find the second order derivatives of the function.

sin (log x)

If y = 5 cos x – 3 sin x, prove that `(d^2y)/(dx^2) + y = 0`

If y = cos^–1 x, Find `(d^2y)/dx^2` in terms of y alone.

If y = 3 cos (log x) + 4 sin (log x), show that x² y₂ + xy₁ + y = 0

If y = Ae^mx + Be^nx, show that `(d^2y)/dx^2  - (m+ n) (dy)/dx + mny = 0`

If y = 500e^7x + 600e^–7x, show that `(d^2y)/(dx^2) = 49y`

If e^y (x + 1) = 1, show that  `(d^2y)/(dx^2) =((dy)/(dx))^2`

If y = (tan^–1 x)², show that (x² + 1)² y₂ + 2x (x² + 1) y₁ = 2

Find the second order derivatives of the function. `x^20`

Verify Rolle’s theorem for the function f (x) = x² + 2x – 8, x ∈ [– 4, 2].

Examine if Rolle’s Theorem is applicable to any of the following functions. Can you say some thing about the converse of Rolle’s Theorem from these examples?

f (x) = [x] for x ∈ [5, 9]

Examine if Rolle’s Theorem is applicable to any of the following functions. Can you say some thing about the converse of Rolle’s Theorem from these examples?

f (x) = [x] for x ∈ [– 2, 2]

Examine if Rolle’s Theorem is applicable to any of the following functions. Can you say some thing about the converse of Rolle’s Theorem from these examples?

f (x) = x² – 1 for x ∈ [1, 2]

If f : [– 5, 5] → R is a differentiable function and if f ′(x) does not vanish anywhere, then prove that f (– 5) ≠ f (5).

Verify Mean Value Theorem, if f (x) = x² – 4x – 3 in the interval [a, b], where a = 1 and b = 4.

Verify Mean Value Theorem, if f (x) = x³ – 5x² – 3x in the interval [a, b], where a = 1 and b = 3. Find all c ∈ (1, 3) for which f ′(c) = 0.

Examine the applicability of Mean Value Theorem for all three functions given in the above exercise 2. 

Differentiate w.r.t. x the function (3x2 – 9x + 5)9

Differentiate w.r.t. x the function sin³ x + cos⁶ x

Differentiate w.r.t. x the function (5x)^{3cos 2x}

Differentiate w.r.t. x the function `sin^(–1)(xsqrtx ), 0 ≤ x ≤ 1`

Differentiate w.r.t. x the function `(cos^(-1) x/2)/sqrt(2x+7), -2 < x < 2`

Differentiate w.r.t. x the function `cot^(-1) [(sqrt(1+sinx) + sqrt(1-sinx))/(sqrt(1+sinx) - sqrt(1-sinx))]`, ` 0 < x < pi/2`

Differentiate w.r.t. x the function (log x)^{log x}, x > 1

Differentiate w.r.t. x the function cos (a cos x + b sin x), for some constant a and b.

Differentiate w.r.t. x the function (sin x – cos x) ^{(sin x – cos x)}, `pi/4 < x < (3pi)/4`

Differentiate w.r.t. x the function x^x + x^a + a^x + a^a, for some fixed a > 0 and x > 0

Differentiate w.r.t. x the function `x^(x^2 -3) + (x -3)^(x^2)`, for x > 3

Find `dy/dx` ,if y = 12 (1 – cos t), x = 10 (t – sin t), `-pi/2< t< pi/2`

Find `dy/dx` , if y = sin^–1 x + sin^–1 `sqrt(1-x^2)`, 0 < x < 1

if `xsqrt(1+y) + ysqrt(1+x) = 0`, for, −1 < x <1, prove that `dy/dx = 1/(1+ x)^2`

If (x – a)² + (y – b)² = c², for some c > 0, prove that

`[1+ (dy/dx)^2]^(3/2)/((d^2y)/dx^2)` is a constant independent of a and b.

If cos y = x cos (a + y), with cos a ≠ ± 1, prove that `dy/dx = cos^2(a+y)/(sin a)`

If x = a (cos t + t sin t) and y = a (sin t – t cos t), find `(d^2y)/dx^2`

If f (x) = |x|³, show that f ″(x) exists for all real x and find it.

Using mathematical induction prove that  `d/(dx) (x^n) = nx^(n -1)` for all positive integers n.

Using the fact that sin (A + B) = sin A cos B + cos A sin B and the differentiation, obtain the sum formula for cosines

Does there exist a function which is continuos everywhere but not differentiable at exactly two points? Justify your answer ?

if y = `[(f(x), g(x), h(x)),(l, m,n),(a,b,c)]`, prove that `dy/dx` =`|(f'(x), g'(x), h'(x)),(l,m, n),(a,b,c)|`

if `y = e^(acos^(-1)x)`, -1 <= x <= 1 show that `(1- x^2) (d^2y)/(dx^2) -x dy/dx - a^2y = 0`