March 28, 2017

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I am unsure of how to take the derivative of this equation. It may be the exponents that are giving me trouble but I'm not sure exactly.

Find the equation of the tangent line to the curve 4e^xy = 2x + y at point (0,4).

On the left side, is the "xy" the exponent of e, or do you mean 4(e^x)y ?

Use implicit differentiation to get the slope m = dy/dx at x=0, y=4.
If 4 e^(xy) = 2x + y, then
4 e^(xy)* (y + x dy/dx) = 2 + dy/dx
Solve for dy/dx = m. The tangent line equation is then
(y-4) = m x

the exponent of e is "xy".

It's must clearer now, thank you.

In that case, my derivation should be correct. Substitute x=0, y=4 in the equation
4 e^(xy)* (y + x dy/dx) = 2 + dy/dx
4*1*(4 + 0) = 2 + dy/dx
dy/dx = 14
y-4 = 14 x
y = 14 x + 4

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