Question

Let y(x²+100) = 200 (Witch of Agnesi). Complete parts a through d.

a. Use implicit differentiation to find dy/dx.

Answer 1

To find dy/dx using implicit differentiation, apply the product rule to y(x²+100) = 200. Differentiate both sides with respect to x, keeping y implicit, and solve for dy/dx to obtain dy/dx = -y(2x)/(x²+100).

Step-by-step explanation:

To find dy/dx using implicit differentiation for the given equation, y(x²+100) = 200, we need to apply the product rule. The product rule states that if you have a product of two functions, u(x) and v(x), the derivative of this product with respect to x is u'(x)v(x) + u(x)v'(x).

To differentiate the given product y(x²+100), we'll denote y as a function of x, i.e., y(x), and use the product rule:

First, differentiate y as a function of x with respect to x and multiply by the second term (x²+100), which gives us dy/dx(x²+100).

Second, keep y as it is and differentiate the second term (x²+100) with respect to x, which gives us y(2x).

Now we can combine these two derivatives to find the derivative of the left side of the equation with respect to x:

dy/dx(x²+100) + y(2x)

The derivative of the right side of the equation, 200, with respect to x is 0 since it is a constant. So we now have:

dy/dx(x²+100) + y(2x) = 0

To find dy/dx, we solve for dy/dx in the equation:

dy/dx = -y(2x)/(x²+100)