Derivatives and Integrals

A table of derivatives of power, exponential, logarithmic and trigonometric functions, the rules for differentiating sums, products, quotients and composite functions, the tangent line equation and the Newton–Leibniz formula for definite integrals.

27 formulas · Grades 10–11

★ Favourites
All 27 formulas on this page
  1. Derivative of a constant
  2. Derivative of a power function
  3. Derivatives of trigonometric functions
  4. Derivative of an exponential function
  5. Derivative of a product
  6. Derivative of a quotient
  7. Newton–Leibniz formula (integral)
  8. Chain rule (derivative of a composite function)
  9. Derivative of a sum of functions
  10. Equation of a tangent line to a function's graph
  11. Antiderivative of a power function
  12. Derivative of x
  13. Derivative of 1/x
  14. Derivative of a square root
  15. Derivative of tangent
  16. Derivative of cotangent
  17. Derivative of a logarithm with base a
  18. Derivative of the natural logarithm
  19. Antiderivative of a constant
  20. Antiderivative of 1/x
  21. Antiderivative of 1/√x
  22. Antiderivative of sine
  23. Antiderivative of cosine
  24. Antiderivative of 1/cos²x
  25. Antiderivative of 1/sin²x
  26. Antiderivative of an exponential function
  27. Antiderivative of eˣ

Derivative of a constant

Grade 10
(c)′=0(c)' = 0

The derivative of a constant is always zero.

Legend:
  • c — Number c

Derivative of a power function

Grade 11
(xn)′=n⋅xn−1(x^n)' = n \cdot x^{n-1}

Value of the derivative of x^n at a given point x.

Legend:
  • x — Value x
  • n — Exponent n

Derivatives of trigonometric functions

Grade 11
(sin⁡x)′=cos⁡x,(cos⁡x)′=−sin⁡x(\sin x)' = \cos x, \quad (\cos x)' = -\sin x

Calculates the values of (sin x)' and (cos x)' at a point x.

Legend:
  • deg — Angle x (in degrees)

Derivative of an exponential function

Grade 11
(ax)′=axln⁡a,(ex)′=ex(a^x)' = a^x \ln a, \quad (e^x)' = e^x

Value of the derivative of a^x at a point x.

Legend:
  • a — Base a (a>0)
  • x — Exponent x

Derivative of a product

Grade 11
(u⋅v)′=u′v+uv′(u \cdot v)' = u'v + uv'

Differentiation rule for the product of two functions u and v.

Legend:
  • u — First function
  • u' — Derivative of the first function
  • v — Second function
  • v' — Derivative of the second function

Derivative of a quotient

Grade 11
(uv)′=u′v−uv′v2\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}

Differentiation rule for the quotient of two functions u and v.

Legend:
  • u — First function
  • u' — Derivative of the first function
  • v — Second function
  • v' — Derivative of the second function

Newton–Leibniz formula (integral)

Grade 11
∫abf(x)dx=F(b)−F(a)\int_a^b f(x)dx = F(b) - F(a)

Calculates a definite integral using values of the antiderivative.

Legend:
  • F(b) — Value of the antiderivative at point b
  • F(a) — Value of the antiderivative at point a

Chain rule (derivative of a composite function)

Grade 11
(f(g(x)))′=f′(g(x))⋅g′(x)(f(g(x)))' = f'(g(x)) \cdot g'(x)

Differentiation rule for a composite function (the chain rule).

Legend:
  • fu — Value f'(u)
  • gx — Value g'(x)

Derivative of a sum of functions

Grade 11
(u+v)′=u′+v′(u+v)' = u' + v'

The derivative of a sum of functions equals the sum of their derivatives.

Legend:
  • up — Derivative u'
  • vp — Derivative v'

Equation of a tangent line to a function's graph

Grade 11
y=f(a)+f′(a)(x−a)y = f(a) + f'(a)(x-a)

Value of the tangent line to a function's graph at a point x.

Legend:
  • fa — Function value f(a)
  • fpa — Derivative value f'(a)
  • a0 — Coordinate x0
  • x — Argument x

Antiderivative of a power function

Grade 11
F(x)=xn+1n+1F(x) = \frac{x^{n+1}}{n+1}

Antiderivative of the power function xⁿ for n ≠ −1.

Legend:
  • x — Value x
  • n — Exponent n

Derivative of x

Grade 11
x′=1x' = 1

The derivative of the variable x with respect to itself is always 1.

Legend:
  • x — Value x

Derivative of 1/x

Grade 11
(1x)′=−1x2\left(\frac{1}{x}\right)' = -\frac{1}{x^2}

Derivative of the inverse-proportionality function at a point x.

Legend:
  • x — Value x

Derivative of a square root

Grade 11
(x)′=12x(\sqrt{x})' = \frac{1}{2\sqrt{x}}

Derivative of the square root function at a point x (x > 0).

Legend:
  • x — Value x

Derivative of tangent

Grade 11
(tg⁡x)′=1cos⁡2x(\operatorname{tg} x)' = \frac{1}{\cos^2 x}

Derivative of the tangent function at a point x.

Legend:
  • deg — Angle x (in degrees)

Derivative of cotangent

Grade 11
(ctg⁡x)′=−1sin⁡2x(\operatorname{ctg} x)' = -\frac{1}{\sin^2 x}

Derivative of the cotangent function at a point x.

Legend:
  • deg — Angle x (in degrees)

Derivative of a logarithm with base a

Grade 11
(log⁡ax)′=1xln⁡a(\log_a x)' = \frac{1}{x \ln a}

Derivative of a logarithmic function with an arbitrary base a.

Legend:
  • x — Value x
  • a — Base a (a>0, a≠1)

Derivative of the natural logarithm

Grade 11
(ln⁡x)′=1x(\ln x)' = \frac{1}{x}

Derivative of the natural logarithm at a point x.

Legend:
  • x — Value x

Antiderivative of a constant

Grade 11
F(x)=kx+CF(x) = kx + C

Antiderivative of a constant k.

Legend:
  • k — Coefficient k
  • x — Value x

Antiderivative of 1/x

Grade 11
F(x)=ln⁡∣x∣+CF(x) = \ln|x| + C

Antiderivative of the inverse-proportionality function.

Legend:
  • x — Value x

Antiderivative of 1/√x

Grade 11
F(x)=2x+CF(x) = 2\sqrt{x} + C

Antiderivative of the reciprocal of the square root of x.

Legend:
  • x — Value x

Antiderivative of sine

Grade 11
F(x)=−cos⁡x+CF(x) = -\cos x + C

Antiderivative of the sine function.

Legend:
  • deg — Angle x (in degrees)

Antiderivative of cosine

Grade 11
F(x)=sin⁡x+CF(x) = \sin x + C

Antiderivative of the cosine function.

Legend:
  • deg — Angle x (in degrees)

Antiderivative of 1/cos²x

Grade 11
F(x)=tg⁡x+CF(x) = \operatorname{tg} x + C

Antiderivative of the reciprocal of cosine squared.

Legend:
  • deg — Angle x (in degrees)

Antiderivative of 1/sin²x

Grade 11
F(x)=−ctg⁡x+CF(x) = -\operatorname{ctg} x + C

Antiderivative of the reciprocal of sine squared.

Legend:
  • deg — Angle x (in degrees)

Antiderivative of an exponential function

Grade 11
F(x)=axln⁡a+CF(x) = \frac{a^x}{\ln a} + C

Antiderivative of an exponential function with an arbitrary base a.

Legend:
  • a — Base a (a>0, a≠1)
  • x — Value x

Antiderivative of eˣ

Grade 11
F(x)=ex+CF(x) = e^x + C

Antiderivative of the exponential function eˣ.

Legend:
  • x — Value x

Related topics