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Math Help - Differentiating both sides of an equation

  1. #1
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    Differentiating both sides of an equation

    Hi guys,

    When is it justified to differentiate both sides of an equation? I'm just reading a textbook, which out of the blue takes the derivative of both sides, without any real justification. For instance, I can't take the derivative of 5x = x^2 + 5x -4, and still preserve that equality.

    Thanks in advance,

    HTale.
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  2. #2
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    Quote Originally Posted by HTale View Post
    Hi guys,

    When is it justified to differentiate both sides of an equation? I'm just reading a textbook, which out of the blue takes the derivative of both sides, without any real justification. For instance, I can't take the derivative of 5x = x^2 + 5x -4, and still preserve that equality.

    Thanks in advance,

    HTale.
    diffferentiation is used to find the instantaneous rate of change.

    if you differentiate the equation 5x = x^2 + 5x - 4 , then you are determining a new equation that tell you where the slope of the two graphs are equal.

    \frac{d}{dx}(5x = x^2 + 5x - 4)

     <br />
5 = 2x + 5<br />

    x = 0 is the location where the slope of y = 5x and the slope of y = x^2 + 5x - 4 are equal.
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  3. #3
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    If two functions f(x) and g(x) are identically equal then you can differentiate both sides of the equation f(x) = g(x), and the resulting equation f'(x) = g'(x) will also be identically true. For example, \sin 2x = 2\sin x\cos x (for all x). Differentiate both sides and you get 2\cos2x = 2\cos^2x-2\sin^2x, also true for all x.

    But the equation 5x = x^2+5x-4 is not an identity. It is only true for two values of x. When you differentiate it you get a completely different equation, which has a different interpretation, as skeeter has pointed out.
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