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    148365
  • 5.2 Implicit differentiation

    We can differentiate any function that is written as a formula, and find a result in terms of a formula. However, sometimes the original problem can't be written in any nice way as a formula. For example, suppose we want to find dy/dx in a case where the relationship between x and y is given by the following equation:

    y7+y = x7+x2 .

    There is no equivalent of the quadratic formula for seventh-order polynomials, so we have no way to solve for one variable in terms of the other in order to differentiate it. However, we can still find dy/dx in terms of x and y. Suppose we let x grow to x+dx. Then for example the x2 term will grow to (x+dx)2=x+2dx+dx2. The squared infinitesimal is negligible, so the increase in x2 was really just 2dx, and we've really just computed the derivative of x2 with respect to x and multiplied it by dx. In symbols,

    eq_4c2ae85d.png

    eq_0284f741.png

    That is, the change in x2 is 2x times the change in x. Doing this to both sides of the original equation, we have

    eq_d62973ed.png

    eq_fc759d0b.png

    eq_636051ad.png

    eq_2920d1b4.png

    This still doesn't give us a formula for the derivative in terms of x alone, but it's not entirely useless. For instance, if we're given a numerical value of x, we can always use Newton's method to find y, and then evaluate the derivative.