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Implicit Differentiation

Not all equations can be written like y = f(x), so taking the derivative can be tricky. Save the mess and do it directly with implicit differentiation.

Rational Functions

         

If \[\frac { x ( x - 6y) } { 5 - 4 y^ 2 } = 1, \] what is \(\displaystyle \frac{dy}{dx}?\)

If \( \frac{ 3 x ^2 + 7 xy^2 } { 8 + 5 y^3 } = 1 \), what is \(\displaystyle \frac{dy}{dx}?\)

If point \((1,a)\) with \(a>0\) lies on the curve \[\frac{y}{x}-\frac{9x}{y}=8,\] what is the derivative \(\displaystyle \frac{dy}{dx}\) of the curve at the point?

Given \[\frac{(x^2+y^2-11x)^2}{x^2+y^2}=4, \] what is the value of \(\displaystyle \frac{dy}{dx}\) when \(x=0\) and \(y=2 ?\)

If \[\frac{6x-y^3}{y+x^2}=x+4, \] what is \(\displaystyle \frac{dy}{dx}?\)

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