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    Definition of the Derivative, Product and Quotient Rules

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    1. Differentiate from first principles( for x radians):

    a) sin x
    b) cos x

    2. Products and quotients
    For a function, f(x), which can be expressed as a product or quotient of other functions, u(x) and v(x), there exist

    a) the product rule, f(x) = u(x) ? v(x),
    df/dx = u ? dv/dx + v ? du/dx, and

    b) the quotient rule, f(x) = u(x)/v(x),
    df/dx =( v? du/dx - u? dv/dx) /v²

    Prove these rules form first principles. It can be assumed that u(x) and v(x) are both differentiable.

    3. Negative powers of x
    Prove form first principles that for f(x) = x&#8319;, where n < 0, df/dx = nx &#8319; &#8254; ¹

    The result for f(x) = x&#8319;, where n > 0, can be assumed; as can the sum, difference, product and quotient rules ( if / as appropriate).

    4. For f(x) = x&#8319;, where n > 0, df/dx = nx. For f(x) =x, where n = 0, f(x) is a constant.

    Which function, f(x), has the derivative df/dx = x&#8254; ¹ = 1/x ?

    Prove this from first principles - from the definition of the derivative.

    © BrainMass Inc. brainmass.com March 4, 2021, 6:15 pm ad1c9bdddf
    https://brainmass.com/math/derivatives/definition-derivative-product-quotient-rules-37311

    Solution Preview

    NOTE: In the following, all limits in symbol of "lim " mean taking limit when h goes to 0.

    1. Differentiate from first principles( for x radians):

    a) sin x

    Proof.

    [sinx]'=lim [sin(x+h)-sinx]/h
    =lim [2cos(x+h/2)sin(h/2)]/h
    =cosx*lim sin(h/2)/(h/2)
    =cosx

    since lim sin(h/2)/(h/2)=1

    b) cos x

    Proof.

    [cosx]'=lim [cos(x+h)-cosx]/h
    =lim [-2sin(x+h/2)sin(h/2)]/h
    =-sinx*lim sin(h/2)/(h/2)
    =-sinx
    since lim sin(h/2)/(h/2)=1

    2. Products and quotients
    For a function, f(x), which can be expressed as a product or quotient of other functions, u(x) and v(x), there exist

    a) the product rule, f(x) = u(x) ? v(x),
    df/dx = u ? dv/dx + v ? du/dx,

    Proof. If f(x) = u(x) ? v(x), then
    df/dx=lim [f(x+h)-f(x)]/h
    =lim [u(x+h)v(x+h)-u(x)v(x)]/h
    =lim [u(x+h)v(x+h)-u(x)v(x+h)+u(x)v(x+h)-u(x)v(x)]/h
    ...

    Solution Summary

    Derivatives are found and the product and quotient rules are proven from first principles. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.

    $2.49

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