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Calculus basics

Quotient rule

Differentiates one function divided by another.

(uv)′=u′v−uv′v2

Calculator

The live calculator needs JavaScript. The worked example below shows the same steps it would.

See it move

−6−4−2246−5510
  • At x = 3: f(x) = 4.5, slope f′(x) = −1.25 (coral tangent). Dashed: the derivative f′.

What the symbols mean

D
the value of the derivative (u/v)′ at the chosen x
a
in the calculator, u=ax+b
b
the constant in u
c
in the calculator, v=cx+d
d
the constant in v
x
the point where the derivative is evaluated
v
the denominator (it must not be 0)

When to use it

Use it for fractions with x on top and bottom: rational functions, tan⁡x=sin⁡x/cos⁡x, exx. Remember "low d-high minus high d-low, over low squared". The calculator takes ax+bcx+d, whose derivative simplifies to ad−bc(cx+d)2.

Worked example

Find the derivative of f(x) = (2x + 3)/(x − 1) at x = 3. Here a = 2, b = 3, c = 1 and d = −1.

  1. Quotient rule: bottom times derivative of top, minus top times derivative of bottom, all over bottom squared.

    (uv)′=u′v−uv′v2

  2. Here u=2⁢x+3, u′=2, v=x−1 and v′=1.

  3. Evaluate at x:

    u=a⁢x+b
    =2·3+3
    =6+3
    =9
  4. v=c⁢x+d
    =1·3+(−1)
    =3+(−1)
    =2

  5. Combine:

    D=u′⁢v−u⁢v′v2
    =2·2−9·122
    =4−94
    =−54
    =−1.25

  6. For this family the numerator is always the constant ad − bc = −5.

Answer f′(3) = −1.25

Why it works

Write u/v=u·v−1 and use the product rule with the chain rule: u′v−1−uv−2v′, then put it over the common denominator v2.

Common mistakes

  • Reversing the subtraction: it is u′v−uv′; the other order flips the sign.
  • Forgetting to square the denominator.
  • Evaluating where the denominator is 0: the function has no derivative there (an asymptote).