The Derivative by Definition

Key Questions

  • How do I find the derivative of a function at a given point?

    First you have to calculate the derivative of the function.

    f(x)=x^3f(x)=x3

    f'(x)=3x^2

    Then if we want to find the derivative of f(x) when x=4 then we substitute that value into f'(x).

    f'(4)=3(4)^2=3*16=48

    AJ Speller · · Oct 1 2014
  • What is the formal definition of a derivative?

    The formal definition of derivative of a function y=f(x) is:
    y'=lim_(Deltax->0)(f(x+Deltax)-f(x))/(Deltax)

    The meaning of this is best understood observing the following diagram:

    enter image source here

    The secant PQ represents the mean rate of change (Deltay)/(Deltax) of your function in the interval between x and x+Deltax.

    If you want the rate of change, say, at P you "move" point Q (and the secant with it) to meet point P as in:
    enter image source here

    In doing so you must reduce Deltax. If Delta x->0 you'll get the tangent in P whose inclination will give the inclination at P and thus the derivative at P.

    enter image source here

    Hope it helps!

    Gió · 1 · May 29 2015
  • How do I find the derivative of a fraction?

    Answer:

    We use quotient rule as described below to differentiate algebraic fractions or any other function written as quotient or fraction of two functions or expressions

    Explanation:

    When we are given a fraction say f(x)=(3-2x-x^2)/(x^2-1). This comprises of two fractions - say one g(x)=3-2x-x^2 in numerator and the other h(x)=x^2-1, in the denominator. Here we use quotient rule as described below.

    Quotient rule states if f(x)=(g(x))/(h(x))

    then (df)/(dx)=((dg)/(dx)xxh(x)-(dh)/(dx)xxg(x))/(h(x))^2

    Here g(x)=3-2x-x^2 and hence (dg)/(dx)=-2-2x and as h(x)=x^2-1, we have (dh)/(dx)=2x and hence

    (df)/(dx)=((-2-2x)xx(x^2-1)-2x xx(3-2x-x^2))/(x^2-1)^2

    = (-2x^3-2x^2+2x+2-6x+4x^2+2x^3)/(x^2-1)^2

    = (2x^2-4x+2)/(x^2-1)^2

    or (2(x-1)^2)/(x^2-1)^2

    = 2/(x+1)^2

    Observe that (3-2x-x^2)/(x^2-1)=((1-x)(3+x))/((x+1)(x-1))=(-3-x)/(x+1) and using quotient rule

    (df)/(dx)=(-(x+1)-(-3-x))/(x+1)^2=2/(x+1)^2

    Shwetank Mauria · 2 · May 1 2018

Questions

Limits, Motion, and the Tangent Line