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.
Just differtiate each term with respect to x, remembering that y is a function of x. You often end up with an expression for y' that involves both x and y, but is is a valid equation. You may have to evaluate y before you can calculate y'. Here is what you get:
x*dy/dx + y + 2 + 6x = 0
dy/dx = -(y + 2 + 6x)/x
= -(y+2)/x -6
thank you
Sure! I can help you understand how to solve this problem using implicit differentiation.
To begin solving this problem, we need to differentiate both sides of the equation with respect to x. The left-hand side differentiates as follows:
d(xy+2x+3x^2)/dx
Now we need to apply the product rule to the term xy. The product rule states that if u = xy, then du/dx = x(dy/dx) + y(dx/dx). Applying the product rule to xy, we get:
x(dy/dx) + y(dx/dx) + (d2x/dx)(y) + (dx/dx)(2x) + (d3x^2/dx)
Simplifying this expression, we have:
x(dy/dx) + y(1) + 0 + 2x + 6x
Which can be further simplified as:
x(dy/dx) + y + 8x
Now, let's differentiate the right-hand side of the equation, which is simply 4. Differentiating a constant gives us 0.
So, the differentiation is:
x(dy/dx) + y + 8x = 0
Now, we want to find dy/dx, which represents the derivative of y with respect to x. To isolate dy/dx, we need to rearrange the equation:
x(dy/dx) + y = -8x
Now, we can solve for dy/dx by dividing both sides by x:
dy/dx = (-8x - y) / x
And that's the derivative of y with respect to x, obtained through implicit differentiation. In this case, the derivative is given in terms of x and y, which means it's implicit and not explicit.
Now, if you want to find the derivative at a specific point (x0, y0), you can substitute these values into the derivative expression to get the numerical value.
I hope this explanation helps you understand how to solve problems using implicit differentiation. Let me know if you have any further questions!