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June 6th, 2017, 04:15 AM   #1
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Difference between partial and implicit differentiation

Hi guys the question is:
Find the equations of the tangent and normal lines to the curve 3x^2-14xy+8y^2-25=0
at the point P = (−1, 1).

would i use partial or implicit differentiation here? Thanks !
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June 6th, 2017, 04:46 AM   #2
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Use implicit differentiation to determine $\dfrac{dy}{dx}$, then evaluate at (-1,1) to find the tangent slope ... opposite reciprocal of that value to get the normal slope.

Equations of both lines may be found using the point-slope form.
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June 6th, 2017, 09:50 AM   #3
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"Partial differentiation" is used when you have a function, say, f, of two independent variables, x and y. "Implicit differentiation" is used when you have a formula in two variable, such as x and y, and one variable, y, is a function of the other. Here, you are not given a function of x and y but, rather, an expression involving x and y, and you are thinking of y as a function of x.

so, differentiating with respect to x, . Then so .

You can use "partial differentiation" to get that result by thinking of "f" as a function of x and y- . If, also, you think of x and y as functions of some third variable, u, using the chain rule, . Since f(x, y) was a constant, its derivative with respect to any variable is 0: .

Now, take u= x so that dx/du= dx/dx= 1 and dy/du= dy/dx. That previous equation becomes .

From that as before.
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