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Math Help - Back to Basics - What actually is differentiation?

  1. #1
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    Back to Basics - What actually is differentiation?

    First post so be gentle with me

    OK. What actually is differentiation?

    Here's the reason I'm asking this.

    Is it working out the gradient of the tangent of a curve as per my understanding?

    If so, what does it mean to have a tangent (dy/dx solution) which is a quadratic or a polynomial rather than a straight line?

    Also, what is the second derivative actually telling you?

    What about third derivates and beyond?

    Any answers to these would be gratefully received.
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  2. #2
    MHF Contributor
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    I am not good at definitions with Math (I like doing with numbers) but here is what I can say.

    One of the uses of differentiation is to find the first derivative of a function. Then one of the meanings of this first derivative is its value is the gradient or slope of the tangent line to the curve of the function.

    This first derivative may not be linear always (it is linear only if the function is of order 2). As you said, sometimes dy/dx is quadratic (if the function is of order 3) or polynomial.
    "...rather than a straight line"

    Uh, the dy/dx is not the equation of the tangent line. Rather, it is the expression of the value of the gradient of the tangent line only. So dy/dx can be quadratic or polynomial.
    To get the equation of the tangent line, you need the value of dy/dx at the
    point of tangency, and at least the point of tangency, and use the point-slope form of a line.

    ----------------------
    The second derivative is the rate of change of the first derivative. It is the rate of change of the slope of the tangent lines.

    Zeez, the books can explain more or better.

    Third derivative? Rate of change of the 2nd derivative.

    Later on in your studies, you'd understand why these derivatives. Relax.
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  3. #3
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    Thanks for this.

    I strongly suspect I am confusing gradient equation with the tangent equation itself.
    Of course, the gradient equation will have a different numerical result for different values of x which you would then use to find the tangent equation.

    What has confused me is seeing the graph of the first and subsequent derivatives. This has certainly confused my understanding of exactly what I being shown.

    In light of this, the desciption of the second derivative makes sense.
    Further derivatives will probably make more sense when viewed from practical implementations of them.

    Thanks for your help on this.
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