Page:Calculus Made Easy.pdf/87

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CHAPTER IX.

INTRODUCING A USEFUL DODGE.

Sometimes one is stumped by finding that the expression to be differentiated is too complicated to tackle directly.

Thus, the equation

$y=(x^{2}+a^{2})^{\tfrac {3}{2}}$

is awkward to a beginner.

Now the dodge to turn the difficulty is this: Write some symbol, such as $u$ , for the expression $x^{2}+a^{2}$ ; then the equation becomes

$y=u^{\tfrac {3}{2}}$ ,

which you can easily manage; for

${\frac {dy}{du}}={\frac {3}{2}}u^{\tfrac {1}{2}}$ .

Then tackle the expression

$u=x^{2}+a^{2}$ ,

and differentiate it with respect to $x$ ,

${\frac {du}{dx}}=2x$ .

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