Limits: Derivatives of Polynomials
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Introduction

    This lesson page will inform you how to find a derivative of a function using the definition of a derivative. Here are the sections within this page:

    In order to understand derivatives, you must understand several concepts and skills. Review any of these lessons in order to prepare for derivatives.

    ideo: Calculating Slope
    esson: Functions
    esson: Limits



    The derivative of a function refers to the slope of a tangent line of a function at a certain point. Here is a graphical representation of a tangent line for three different parabolas.

geometrical geometry derivative

    To fully understand derivatives, we need to look at its definition.


    The definition of a derivative is a blend of pre-calculus and algebra. Here it is for any function f(x).

definition derivative

    If you would like to learn where this definition comes from, watch this video.

    ideo: Derivation of a Derivative


    Using the definition of a derivative, find the derivative of f(x), given below.

quadratic function

    To get started we write the definition of a derivative.

definition derivative

    Now, we substitute x + h into the function f(x) and subtract f(x) from it.

algebra derivative

    Notice how parentheses were used for substitution. This is extremely important.

    Also notice what happens if we were to allow the h-value to go to zero.

algebra derivative zero denominator

    We would have a zero in the denominator, which is illogical. So, we have to employ other – more algebraic – methods for finding this limit.

    Let’s square the x + h, using algebra instead of letting h go to zero.

algebra derivative

    Next we need to drop the parentheses and use the distributive property, where necessary.

algebra derivative distributive property

    We can simplify the numerator by cleaning up terms.

algebra derivative simplification simplify

    Next, we can simplify the numerator by using algebra.

algebra derivative simplify

    Since we no longer have h in a denominator, let h go to zero.

algebra derivative

    This means…

derivative linear

    It means the slope of the tangent line is a function dependent upon the location of the point on the function. The slope of the tangent line changes depending on where we look on the function.

    If we were interested in finding the slope of the tangent line at x = 3, we would do this…

slope derivative evaluate

    Here is the function (in blue) and the tangent line (in black) at x = 3.

slope derivative evaluate

    If we wanted to know the slope of the tangent line at x = -1, we would do this…

algebra derivative

    ideo: Derivative of a Polynomial
    uiz: Derivatives of Polynomials


    Let’s find the derivative of this function using the definition of a derivative.

linear function

    Here is the definition of a derivative.

definition derivative

    We need to substitute x + h into the function and subtract the function from it.

algebra derivative

    Next, use the distributive property to drop the parentheses.

algebra derivative distributive property

    We can clean up some terms.

algebra derivative simplification

    Next, simplify the expression.

algebra derivative simplification

    Allowing h to go to zero is peculiar for this problem because there are no more h’s that remain. So, allowing h to go to zero does not change the result. The result is simply -4.

derivative constant

    This may seem strange until we look at the original function, f(x).

linear function

    The function f(x) is a line. It has a y-intercept of 5 and a slope of -4. That means everywhere along the line the line has a slope of -4. This is exactly what the derivative is telling us. The derivative tells us the slope of the tangent line is -4, regardless of where we look on the line. Here is the graph of f(x) and a tangent line to the function.

graph graphical derivative

    ideo: Derivative of a Polynomial
    uiz: Derivatives of Polynomials


    Try this instructional video to learn this lesson.

    ideo: Derivative of a Polynomial


    Try this interactive quizmaster to learn this lesson.

    uiz: Derivatives of Polynomials


    Try these lessons, which are related to the sections above.

    esson: Limits
    esson: Limits by Factoring
    esson: Continuity