What is the derivative of $y = \frac{1}{2} (x^{2} - x^{- 2})$ $y=1/2(x^2-x^-2)$ ?

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What is the derivative of $y = \frac{1}{2} (x^{2} - x^{- 2})$ $y=1/2(x^2-x^-2)$ ?

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Answer 1 · 2016-05-21T15:00:10

$x + x^{- 3} = x + \frac{1}{x^{3}}$ $x+x^-3=x+1/x^3$

Explanation:

The first step here is to multiply out the brackets.

$\Rightarrow \frac{1}{2} (x^{2} - x^{- 2}) = \frac{1}{2} x^{2} - \frac{1}{2} x^{- 2}$ $rArr1/2(x^2-x^-2)=1/2x^2-1/2x^-2$

Now both terms can be differentiated using the $power rule$ $color(blue)" power rule"$

$\frac{d}{d x} (a x^{n}) = n a x^{n - 1}$ $d/dx(ax^n)=nax^(n-1)$

$\Rightarrow \frac{d y}{d x} = 2 \times \frac{1}{2} x^{2 - 1} - (- 2) \times \frac{1}{2} x^{- 2 - 1}$ $rArrdy/dx=2xx1/2x^(2-1)-(-2)xx1/2x^(-2-1)$

which simplifies to.

$\frac{d y}{d x} = x + x^{- 3} = x + \frac{1}{x^{3}}$ $dy/dx=x+x^-3=x+1/x^3$

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What is the derivative of $y = \frac{1}{2} (x^{2} - x^{- 2})$ $y=1/2(x^2-x^-2)$ ?

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What is the derivative of $y = \frac{1}{2} (x^{2} - x^{- 2})$ $y=1/2(x^2-x^-2)$ ?