Implicit Differentiation

Functions hiding in equations

The general procedure

Higher derivatives implicitly

Curves with no closed form

In implicit differentiation of x^2 + y^2 = 25, the derivative of y^2 with respect to x is 2y dy/dx. Why does the factor dy/dx appear?

Using implicit differentiation, what is the slope dy/dx of the circle x^2 + y^2 = 25 at the point (3, 4)?

For the circle x^2 + y^2 = 25, the derivative dy/dx = -x/y is undefined where y = 0. What geometric feature occurs at those points?

Differentiating the term xy in the equation x^2 + xy + y^2 = 7, which expression results?

The second derivative of the circle's upper semicircle works out to -25/y^3, which is negative since y is positive up there. What does the negative sign confirm geometrically?

For the circle x^2 + y^2 = 1 with derivative dy/dx = -x/y, what is the tangent slope at the point (0, -1) on the bottom?

Differentiating xy = 1 implicitly gives a relation between dy/dx, x, and y. What is dy/dx? Answer like -y/x.

For x^2 + y^2 = 1, the derivative is dy/dx = -x/y. At the point (0, 1) on top, what is the slope? Answer with a number.

In the equation x^2 + xy = 4, implicit differentiation produces one term from x^2, one from y in the product, and one from x dy/dx. How many total terms before solving? Answer with a number.

Implicit differentiation of x^2/9 + y^2/4 = 1 gives dy/dx equal to a constant times x/y. What is that constant, as a fraction? Answer like -4/9.