Equation of tangent - Implicit or Partial DifferentiatioN?

[dan38]

Homework Statement

Need to find the tangent to the curve at: e^(xy) + x^2*y - (y-x)^2 + 3

I just implicitly differentiate the expression to find the gradient and then use the points given to find the equation, right?
Or does this involve partial differentiation?

Homework Equations

The Attempt at a Solution

 

[Infinitum]

None of those

Just differentiate with respect to x, using the chain rule and product rule.

 

[ehild]

Homework Statement

Need to find the tangent to the curve at: e^(xy) + x^2*y - (y-x)^2 + 3

Is it a curve e^(xy) + x^2*y - (y-x)^2 + 3=constant, or a surface F(x,y)=e^(xy) + x^2*y - (y-x)^2 + 3? And at which point do you need to find the tangent or gradient?

ehild

 

[dan38]

curve at (0,2)
how do you differentiate with respects to "x" if there's a "y" without one of those methods..

 

[Infinitum]

curve at (0,2)
how do you differentiate with respects to "x" if there's a "y" without one of those methods..

As I said, use product and chain rules. For example, the derivative of the first term would be,

$$\frac{de^{xy}}{dx} = e^{xy} (y + x\cdot \frac{dy}{dx})$$

 

[dan38]

ah I see
would it have been wrong to implicit differentiation then?

 

[Infinitum]

ah I see
would it have been wrong to implicit differentiation then?

Implicit differentiation is used when you cannot differentiate the terms using these methods, specifically when you have to integrate function raised to another function of the same variable. This is not the case with the given curve, so there is no need for implicit differentiation.

 

[ehild]

Infinitum, what you did, it is implicit differentiation.

y is given as a function of x implicitly by an equation R(x, y) = 0. We differentiate R(x, y) with respect to x and then with respect to y, multiplying it with dy/dx.

ehild

 

[HallsofIvy]

Implicit differentiation is used when you cannot differentiate the terms using these methods, specifically when you have to integrate function raised to another function of the same variable. This is not the case with the given curve, so there is no need for implicit differentiation.

What, exactly, do you mean by "implicit differentiation"? It is clearly called for in this problem.

 

[Infinitum]

Infinitum, what you did, it is implicit differentiation.

y is given as a function of x implicitly by an equation R(x, y) = 0. We differentiate R(x, y) with respect to x and then with respect to y, multiplying it with dy/dx.

ehild

You're right. I don't know, but absent-mindedly, I interchange definitions sometimes...

Apologies, dan38!