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June 6th, 2017, 04:15 AM  #1 
Member Joined: Jan 2016 From: Blackpool Posts: 44 Thanks: 1  Difference between partial and implicit differentiation
Hi guys the question is: Find the equations of the tangent and normal lines to the curve 3x^214xy+8y^225=0 at the point P = (−1, 1). would i use partial or implicit differentiation here? Thanks ! 
June 6th, 2017, 04:46 AM  #2 
Math Team Joined: Jul 2011 From: Texas Posts: 2,639 Thanks: 1319 
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 pointslope form. 
June 6th, 2017, 09:50 AM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,719 Thanks: 699 
"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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