I do not understand implicit differentiation. One of the problems are: find y' by implicit differentiation xy+2x+3x^2=4.
I would appreciate any help that can be offered.
if you can do the basic rules like the product rule, then you can do implicit differentiating.
xy+2x+3x^2=4
Let me explain term by term
xy is a product, so when you differentiate
you get
x(dy/dx) + y(dx/dx)
or
x(dy/dx) + y since dx/dx = 1
2x would give you 2 and
3x^2 would give you 6x (really it was 6x(dx/dx) which is 6x(1) or 6x)
and of course the derivative of 4 is zero
so you would have
x(dy/dx) + y + 2 + 6x = 0
solving this for dy/dx you would finally have
dy/dx = (-6x - y - 2)/x
You will get questions where there is a dy/dx in several terms.
In that case bring all those terms to one side of your equation, factor out the dy/dx, and solve for dy/dx that way.
the key this is to remember:
if you differentiate an x term you also get a dx/dx hanging around, which is 1, so you don't have to write it, but
if you differentiate a y term you would get a dy/dx hanging around.
Sure! I can help you understand implicit differentiation and how to apply it to this problem.
Implicit differentiation is a technique used to differentiate equations where the dependent variable is not explicitly in terms of the independent variable. In other words, it allows us to find the derivative of variables that are implicitly defined in an equation.
To start, we differentiate both sides of the equation with respect to the independent variable (usually denoted as x). Let's go through the steps for the given problem:
Step 1: Write down the equation: xy + 2x + 3x^2 = 4.
Step 2: Differentiate both sides of the equation with respect to x. Treat y as a function of x and use the product rule for the term xy:
(d/dx)(xy) + (d/dx)(2x) + (d/dx)(3x^2) = (d/dx)(4).
Step 3: Apply the product rule for differentiating xy. The product rule states that for functions u(x) and v(x), (u(x)v(x))' = u'(x)v(x) + u(x)v'(x). In this case, u(x) = x and v(x) = y, so we have:
x(dy/dx) + y(dx/dx) + 2 + 6x = 0.
Step 4: Simplify the equation by simplifying dx/dx and rearranging terms:
x(dy/dx) + y + 2 + 6x = 0.
Step 5: Solve for dy/dx by isolating it on one side of the equation:
x(dy/dx) = -y - 2 - 6x.
dy/dx = (-y - 2 - 6x)/x.
And there you have it! The derivative of y with respect to x (dy/dx) is given by the expression (-y - 2 - 6x)/x.
Note: Sometimes, the equation obtained may be further simplified or rearranged to make it more readable or convenient for further analysis, but the above steps outline the core process of implicit differentiation.
I hope this explanation helps you better understand implicit differentiation and how to apply it to this specific problem.