Stuck on this: A curve has the equation $\displaystyle x^2 - 4xy = y - 7x$. Find the gradient at the point (2,2).

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- Feb 2nd 2009, 11:26 AMBeardImplicit differentiation
Stuck on this: A curve has the equation $\displaystyle x^2 - 4xy = y - 7x$. Find the gradient at the point (2,2).

- Feb 2nd 2009, 12:46 PMRincewind
You can differentiate both sides with respect to x remembering that y is a function of x.

$\displaystyle 2x - 4y - 4xy' = y' - 7$

(N.B> I used the product rule on the second term $\displaystyle (4xy)' = 4y + 4xy'$)

Simplifying you get

$\displaystyle y' = \frac{7+2x-4y}{4x+1}$

Now just substitute $\displaystyle x=2$, $\displaystyle y=2$ and you have the value of $\displaystyle y'$, which is the gradient.

HTH