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I'd like to differentiate $y=x^{\large\frac{-3}{2}}$. So,

$$y + dy = (x + dx)^{\large\frac{-3}{2}} = x^{\large\frac{-3}{2}}\Big(1 + \frac{dx}{x}\Big)^{\large\frac{-3}{2}}$$

is at least a start. How do I calculate the parentheses?

dash
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    I think you are overly complicating things. Please see the rules of differentiating first – Shailesh Sep 06 '15 at 11:29
  • the general rule is $y=x^a \Rightarrow y'=an^{a-1}$. Do you want know how apply this rule ( that is easy) or you want to find the derivative directly from the definition? – Emilio Novati Sep 06 '15 at 11:31
  • I'd like to do it from the definition. (I know that rule). – dash Sep 06 '15 at 11:32
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    I think the OP is probably trying to find the derivative from first principles-i.e. Formal definition of a derivative. – John_dydx Sep 06 '15 at 11:35

2 Answers2

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Try squaring both sides and then eliminating less significant ${dy}^2$, ${dx}^2$, $dxdy$ and ${dx}^3$ etc. kind of terms.
Then you will get it.$$(y+dy)^2 = \frac{1}{(x+dx)^3}$$
$$\implies y^2+{dy}^2+2y\cdot dy = \frac{1}{x^3+{dx}^3+3x^2\cdot dx+3x\cdot {dx}^2}$$
Eliminating less significant higher degrees of $dx$ and $dy$ $$(y^2+2y\cdot dy)(x^3+3x^2\cdot dx) = 1$$
$$\implies x^3y^2+2x^3y\cdot dy+3x^2y^2\cdot dx = 1$$
Using $y = x^{-3/2}$ here, it becomes $$ 1+2x^{3/2}\cdot dy+3x^{-1}\cdot dx=1$$
$$\implies \frac{dy}{dx}=-\frac{3}{2}{x^{-5/2}}$$
Hope that helps.

Lonidard
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Do you know the Binomial theorem?

Expand upto two terms

$$y + dy = (x + dx)^{\large\frac{-3}{2}} = x^{\large\frac{-3}{2}}\Big(1 + \frac{dx}{x}\Big)^{\large\frac{-3}{2}} \approx x^{\large\frac{-3}{2}}\Big(1 + \frac{(-3/2) dx}{x}\Big)^{\large1} $$

Cancel out first term, divide by $x$ and simplify to find

$$ \dfrac{dy}{dx}= -\dfrac32 x ^{-5/2}. $$

Narasimham
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  • Yes but only for positive integers. I guess I need to learn how to apply it in other cases as well. – dash Sep 06 '15 at 11:52
  • The infinite power series for non integral exponent can be used $$ (1+x)^n = 1 + n x + \frac{n(n-1)}{2!}x^2 + ... $$ – Narasimham Sep 06 '15 at 11:56