Fundamental Differentiation Rules and Calculus Exercises

Fundamental Differentiation Rules for Basic Functions

  • Derivative of a constant: (c)′=0(c)' = 0 (where cc is a constant).

  • Derivative of the identity function: (x)′=1(x)' = 1.

  • Power rule for natural numbers: (xn)′=n×xn−1(x^n)' = n \times x^{n-1} where n∈N,n>1n \in \mathbb{N}, n > 1.

  • Derivative of the square root function: (x)′=12x(\sqrt{x})' = \frac{1}{2\sqrt{x}} (defined for x>0x > 0).

  • Derivative of the reciprocal function: (1x)′=−1x2(\frac{1}{x})' = -\frac{1}{x^2}.

Rules for Operations on Functions

  • Scalar Multiplication Rule: [K⋅f(x)]′=K⋅f′(x)[K \cdot f(x)]' = K \cdot f'(x).

  • Sum Rule: (u+v)′=u′+v′(u + v)' = u' + v' (where uu and vv are functions differentiable at point xx).

  • Difference Rule: (u−v)′=u′−v′(u - v)' = u' - v' (where uu and vv are functions differentiable at point xx).

  • Product Rule: (u⋅v)′=u′⋅v+v′⋅u(u \cdot v)' = u' \cdot v + v' \cdot u.

  • Quotient Rule: (uv)′=u′⋅v−v′⋅uv2(\frac{u}{v})' = \frac{u' \cdot v - v' \cdot u}{v^2}.

Derivatives of Trigonometric Functions

  • Sine function: (sin⁡(x))′=cos⁡(x)(\sin(x))' = \cos(x).

  • Cosine function: (cos⁡(x))′=−sin⁡(x)(\cos(x))' = -\sin(x).

  • Tangent function: (tan⁡(x))′=1cos⁡2(x)(\tan(x))' = \frac{1}{\cos^2(x)}.

  • Cotangent function: (cot⁡(x))′=−1sin⁡2(x)(\cot(x))' = -\frac{1}{\sin^2(x)}.

Differentiation Exercises: Basic and Polynomial Functions

Câu 1: Find the derivative of the following functions:

  • a) y=x5y = x^5

  • b) y=x3+2x2y = x^3 + 2x^2

  • c) y=x2023+x+2023y = x^2023 + x + 2023

  • d) y=x10+x2+x+1y = x^{10} + x^2 + x + 1

  • e) y=1x+2023y = \frac{1}{x + 2023}

Câu 2: Find the derivative of the following functions:

  • a) y=4x2y = 4x^2

  • b) y=x82y = \frac{x^8}{2}

  • c) y=x6+x3+x2+2023y = x^6 + x^3 + x^2 + 2023

  • d) y=x44y = \frac{x^4}{4}

  • e) y=x55+x33+x5y = \frac{x^5}{5} + \frac{x^3}{3} + \frac{x}{5}

  • f) y=x42+x33+x24+15y = \frac{x^4}{2} + \frac{x^3}{3} + \frac{x^2}{4} + \frac{1}{5}

  • g) y=x4−2x2y = \frac{x^4 - 2x}{2}

  • h) Solve the inequality f′(x)>g′(x)f'(x) > g'(x) given that:   f(x)=x33+x22+3xf(x) = \frac{x^3}{3} + \frac{x^2}{2} + 3xg(x)=x33−x22+2xg(x) = \frac{x^3}{3} - \frac{x^2}{2} + 2x

  • i) Prove that for the function f(x)=x55+x33−x22+xf(x) = \frac{x^5}{5} + \frac{x^3}{3} - \frac{x^2}{2} + x, the following identity holds:   f′(1)+f′(−1)=4⋅f′(0)f'(1) + f'(-1) = 4 \cdot f'(0)

Differentiation Exercises: Product Rule and Powers

Câu 3: Find the derivative of the following functions:

  • a) y=(2x+1)(x+3)y = (2x + 1)(x + 3)

  • b) y=(x2+3)(x5−8x3)y = (x^2 + 3)(x^5 - 8x^3)

  • c) y=(x2+2)2y = (x^2 + 2)^2

  • d) y=x⋅(x+2x)y = x \cdot (x + 2\sqrt{x})

  • e) y=(x2+1)(x2+2)(x2+3)y = (x^2 + 1)(x^2 + 2)(x^2 + 3)

Differentiation Exercises: Trigonometric Functions

Câu 4: Find the derivative of the following functions:

  • a) y=x3+sin⁡(x)−3cos⁡(x)+2tan⁡(x)y = x^3 + \sin(x) - 3\cos(x) + 2\tan(x)

  • b) y=1+cos⁡(x)+2cot⁡(x)−2023sin⁡(3)y = 1 + \cos(x) + 2\cot(x) - 2023\sin(3)

  • c) Solve the equation f′(x)=0f'(x) = 0 given that:   f(x)=sin⁡(x)+cos⁡(x)2+xf(x) = \sin(x) + \frac{\cos(x)}{2} + x

Differentiation Exercises: Advanced Products and Quotients

Câu 5: Find the derivative of the following functions:

  • a) y=(2x+1)⋅tan⁡(x)y = (2x + 1) \cdot \tan(x)

  • b) y=(x3+9)(x2+2)(x2+9x+1)y = (x^3 + 9)(x^2 + 2)(x^2 + 9x + 1)

  • c) y=(sin⁡(x)+cos⁡(x)sin⁡(x)−cos⁡(x))2023y = \left( \frac{\sin(x) + \cos(x)}{\sin(x) - \cos(x)} \right)^{2023}

  • d) y=x−5x−2y = \frac{x - 5}{x - 2}

  • e) y=3x2+2x+3x−4y = \frac{3x^2 + 2x + 3}{x - 4}

  • f) y=sin⁡(x)−cos⁡(x)sin⁡(x)+cos⁡(x)y = \frac{\sin(x) - \cos(x)}{\sin(x) + \cos(x)}

  • g) y=sin⁡(x)+cos⁡(x)sin⁡(x)−cos⁡(x)y = \frac{\sin(x) + \cos(x)}{\sin(x) - \cos(x)}

  • h) Solve the inequality f′(x)<0f'(x) < 0 given that:   f(x)=x2−2x+1x+1f(x) = \frac{x^2 - 2x + 1}{x + 1}