Learn – Calculus – Derivatives

Definition

Exercises

Problem set

Find f'(2) for each of the following using the definition of the derivative.

  1. f(x) = 10x
  2. f(x) = 5x+7
  3. f(x) = 2x^2
  4. f(x) = \frac{1}{x}
  5. f(x) = \frac{1}{3x}
  6. f(x) = \frac{1}{x^2}
  7. f(x) = \frac{1}{5x^2}
  8. f(x) = \sqrt{x}
  9. f(x) = 2x^3-3x^2+2x-1

Problem set

Find the value of each of the following assuming f(x) = |x|.

  1. f'(3)
  2. f'(-5)
  3. f'(0)

Problem set

Find \frac{dy}{dx} for each of the following using the definition of the derivative.

  1. y = x^2
  2. y = x^3
  3. y = \frac{3}{x}
  4. y = -\frac{8}{x}
  5. y = \frac{4}{x^2}
  6. y = \frac{1}{3x^2}

Geometric significance and differentiability

Exercises

Problem set

  1. For the function h(x) shown below, estimate the following:
    1. h'(2.5)
    2. h'(-2.5)
    3. h'(0.2)
    4. h'(-0.2)


  2. For the function f(x) shown below, estimate the following:
    1. f'(4)
    2. f'(-4)
    3. f'(0.1)
    4. f'(-0.1)


Problem set

Identify where each of the following functions is differentiable.






Problem set

Identify where each of the following functions is differentiable.

  1. |x|
  2. |x-3|
  3. \log x + \cos x
  4. \lfloor x \rfloor
  5. \lceil x + 0.5 \rceil
  6. x^2 + |x|
  7. |x-1| - |x+5|
  8. \frac{x^2+1}{\sqrt{x}+1}
  9. \sqrt[3]{x}
  10. \sqrt[5]{x^3}

Problem set

Identify where each of the following functions is differentiable.

  1. \frac{x^2+1}{x+1}
  2. \lfloor x \rfloor. (Here, \lfloor x \rfloor denotes the greatest integer that is less than or equal to x. For example, \lfloor 3.12 \rfloor = 3, \lfloor -2.12 \rfloor = -3 and \lfloor -100 \rfloor = -100)
  3. |\sin x|

Problem set

  1. What is the slope of tangent to the graph of 4x^3-7x^2+9x-5 at x=1?
  2. What is the slope of tangent to the graph of \frac{1}{x} at x=2?
  3. What is the slope of tangent to the graph of 3\sqrt{x} at x=9?
  4. What is the slope of tangent to the graph of e^x at x=2?
  5. What is the slope of tangent to the graph of \sin x at x=\frac{\pi}{4}?

Problem set

  1. Give the equation to the tangent to the graph of x^2 at x=3.
  2. Give the equation to the tangent to the graph of 2e^x at x=1.
  3. Give the equation to the tangent to the graph of \tan x at x=\frac{\pi}{4}.

Power rule, Sum rule, product rule and quotient rule

Exercises

Problem set

In each of the following problems, a function f(x) is given. In each case, find f'(x).

  1. x^3
  2. x^9

Problem set

In each of the following problems, a function f(x) is given. In each case, find f'(x).

  1. x^{41}-x^{5}
  2. x^{77}+x^{51}-x^{43}
  3. 10x^{99}-100x^{40}
  4. 7x^{4}-5x^3+11x^2-4x+15
  5. 8x^3-3x^2+12x-99
  6. 11x^5-4x^3+2x-5
  7. x+\sqrt{x}

Problem set

Find the derivative of each of the following functions.

  1. 3e^x
  2. x^2e^x
  3. 3x^7e^x
  4. e^x\sin x
  5. \sin x\cos x
  6. 3x^{10}\cos x

Problem set

Find the derivative of each of the following functions.

  1. x^{50}\left(\sqrt{x}-1\right)
  2. \frac{x^4+x^2}{x+1}
  3. \frac{\sqrt{x}+1}{\sqrt{x}-1}
  4. (\sqrt{x}-1)\frac{x-1}{x+1}

Trigonometric, exponential and logarithmic functions

Exercises

Problem set

Find the derivative of each of the following using the definition of derivative.

  1. \sin x
  2. \cos x

Problem set

Find the derivative of each of the following functions.

  1. \tan x
  2. \sec x
  3. \cot x
  4. \csc x
  5. \log^x_3

Problem set

Find the derivative of each of the following functions.

  1. x\sin x
  2. (2x+1)\sin x
  3. (3x^2-7x+8)\cos x
  4. (7x^5-3x^2-4x-99)\sin x
  5. x^2e^x
  6. \frac{\sin x}{e^x}
  7. \frac{\tan x}{2^x}

Problem set

Prove the following.

  1. \frac{d}{dx} \left(\frac{\sin x}{1+\cos x}\right) = \frac{1}{1+\cos x}
  2. \frac{d}{dx} \left(\frac{\tan x}{1+\sec x}\right) = \frac{\sec x}{1+\sec x}
  3. \frac{d}{dx} \left(\frac{\cot x}{1+\csc x}\right) = -\frac{\csc x}{1+\csc x}
  4. \frac{d}{dx} \left(\frac{\cot x}{1+\cot x}\right) = -\frac{1}{1+\sin 2x}
  5. \frac{d}{dx} \left(\frac{\cos x}{1+\sin x}\right)^3 = \frac{3(\sin x - 1)}{(1+\sin x)^2}

Problem set

Find the derivative of each of the following functions.

  1. \sqrt{x}\sec x\tan x
  2. \frac{\csc x\cot x}{\sqrt{x}}
  3. (x+1)\sin 3x\cos 4x^2
  4. x^22^x\sin x^2

Chain rule

Exercises

Problem set

Find the derivative of each of the following functions.

  1. \sin\left(x^9\right)
  2. \cos(\sin x)
  3. e^{9x}
  4. \cos^5 x
  5. \cos x^5

Problem set

Find the derivative of each of the following functions.

  1. \left(e^x+\sin x\right)^{51}
  2. (3x^2-7)^{100}
  3. (5x^{10}-11x^2+41)^{23}
  4. \cos^2 x^2
  5. \tan^5 x^5

Problem set

Find the derivative of each of the following functions.

  1. \sin 2x \cos 2x
  2. e^{e^x}
  3. 2^{\tan x}
  4. \sqrt{1+3x^3}
  5. \tan \sqrt{x}
  6. \frac{x}{\sqrt{1+2x^2}}

Problem set

Find the derivative of each of the following functions.

  1. e^{\tan x} + \ln (\cot x)
  2. \ln (\ln (\ln x))
  3. 2^{2x^2}
  4. \log^{\sec \sqrt{x}}_2
  5. \log^{\frac{x^2\sqrt{x-5}}{3}}_5
  6. 5^{5^x}

Problem set

Find the derivative of each of the following functions.

  1. 3^{\tan x}
  2. \log^{2x^2}_x
  3. \log^{\sec x}_{\csc x}
  4. \log^{\tan x}_{\cot x}
  5. x^x
  6. \csc^x x
  7. \ln^x x
  8. \arcsin\left(x^4\right)

Problem set

Justify the following.

  1. \frac{d}{dx} x^{\ln x^2} = \frac{2x^{\ln x^2}\ln x}{x}
  2. \frac{d}{dx} x^{\ln \left(\sqrt[3]{x}\right)} = \frac{2x^{\ln \left(\sqrt[3]{x}\right)}\ln x}{3x}

Implicit differentiation

Exercises

Problem set

Find the derivatives \frac{dy}{dx} for the following at the given points.

  1. x^2+y^2=25, at \left(-3,-4\right)
  2. 9(x-3)^2+9(y+3)^2 = 25, at \left(4,-\frac{5}{3}\right)
  3. \tan y = xy, at \left(\frac{4}{\pi},\frac{\pi}{4}\right)
  4. (x-y)^2 = 144, at (11,-1) and (-1, 11)
  5. 2^x\ln y = xy, at (0,1)
  6. xy\csc y = 1, at \left(\frac{2}{\pi},\frac{\pi}{2}\right)

Derivative of inverse of a function

Exercises

Problem set

  1. Given that f'(x) = \frac{1}{\sqrt{2x-1}} and (2,\sqrt{3}) is a point on the graph of f(x), find the value of \left(f^{-1}\right)'(\sqrt{3}).
  2. Given that f(x) = x^7-10x+1 and (1,-8) is a point on the graph of f(x), find the value of \frac {df^{-1}}{dx}(-8).

Problem set

Identify domain and range for each of the following functions, and obtain their derivatives.

  1. \arcsin x
  2. \arccos x
  3. \arctan x
  4. \csc^{-1}(x)
  5. \sec^{-1}(x)
  6. \cot^{-1}(x)

Problem set

Justify the following.

  1. \frac{d}{dx} \arcsin(\cos x) = -1
  2. \frac{d}{dx} \arccos(\sin x) = -1
  3. \frac{d}{dx} \cos(\arcsin(x)) = -\frac{x}{\sqrt{1-x^2}}
  4. \frac{d}{dx} \sec(\arctan(x)) = \frac{x}{\sqrt{1+x^2}}
  5. \frac{d}{dx} \tan(\arcsin(x)) = \frac{1}{(1-x^2)^{3/2}}

Problem set

Justify the following.

  1. \frac{d}{dx} \sin(\arctan(6x)) = \frac{6}{(1+36x^2)^{3/2}}
  2. \frac{d}{dx} \cos(\cot^{-1}(\pi x)) = \frac{\pi}{(1+\pi^2x^2)^{3/2}}
  3. \frac{d}{dx} \sin(\cot^{-1}(3x)) = -\frac{9x}{(1+9x^2)^{3/2}}
  4. \frac{d}{dx} \cos\left(\arctan\left(\frac{x}{2}\right)\right) = -\frac{2x}{(x^2+4)^{3/2}}

Antiderivatives

Exercises

Problem set

Find an antiderivative for each of the following.

  1. x^2
  2. x^5
  3. e^x
  4. \sin x
  5. \cos x
  6. \frac{1}{x}

Problem set

Find an antiderivative for each of the following.

  1. 2x^3-3x^2+5
  2. \frac{1}{x} + \frac{1}{x^2}
  3. \frac{1}{x} + \frac{1}{x^2} + \frac{1}{x^3}
  4. 2e^x
  5. \frac{2^x}{2}
  6. \cos x - \sin x
  7. \sec^2 x