Question

Find the derivative in these points indicated:
x^3 - (2x^2)y + 5x + y - 5 = 0; x=1, y=-1

Answer 1

Show me your best attempt at solving this and I'll help you where you get stuck.

Answer 2

well I know that I have to use the definition of derivatives: \[\lim 0 [ f(Xo+h) - f(Xo)] /h\]... but what if I have a y in it?

Answer 3

Are you sure you have to use the definition of derivatives to solve this implicit differentiation rather than just applying the power, product, and chain rules?

Answer 4

you mean just replace x=1 and y=-1

Answer 5

in the derivative of that function I have? :O

Answer 6

Oh, no not at all. You will have to plug those values into the derivative of that function, not the function itself.

Answer 7

the derivative of that function is: 3x^2 - 4x + 6 = 0 .... am I right?

Answer 8

You are looking to find the derivative, dy/dx, at the single point (1,-1), when
x^3 - (2x^2)y + 5x + y - 5 = 0.

This is called an "implicit function," because it has not been solved for y. Fortunately, we can use "implicit differentiation" to find this derivative, without first solving x^3 - (2x^2)y + 5x + y - 5 = 0 for y.

Let's look at the terms of this relationship one by one:

x^3 is a function of x only, and so we just apply the power rule to find dy/dx here.

(2x^2)y is a product and is itself an implicit function because both x and y appear in the same term. We must use the product rule, and then the chain rule, to find the derivative of this term.

5x is easily differentiated with respect to x.


y is an implicit function of x, and so you must use the power and chain rules to differentiate it.

First, please find use the indicated differentiation rules to differentiate each of these terms. We'll put the results together into a whole when you've finished.

Answer 9

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