Determine \frac{dy}{dx} for the following in terms of x and y, using implicit differentiation:

\displaystyle x^2 + y^2 = 36

Determine \frac{dy}{dx} for the following in terms of x and y, using implicit differentiation:

\displaystyle 15y^2 = 2x^3

Determine \frac{dy}{dx} for the following in terms of x and y, using implicit differentiation:

\displaystyle 3xy^2 + y^3 = 8

Determine \frac{dy}{dx} for the following in terms of x and y, using implicit differentiation:

\displaystyle 9x^2-16y^2 = - 144

Determine \frac{dy}{dx} for the following in terms of x and y, using implicit differentiation:

\displaystyle \frac{x^2}{16} + \frac{3y^2}{13} = 1

Determine \frac{dy}{dx} for the following in terms of x and y, using implicit differentiation:

\displaystyle x^2 + y^2 + 5y = 10

For the relation, determine the equation of the tangent at the given point.

\displaystyle x^2 + y^2 = 13, (2, -3))

For the relation, determine the equation of the tangent at the given point.

\displaystyle x^2 + 4y^2 = 100, (-8, 3)

For the relation, determine the equation of the tangent at the given point.

\displaystyle \frac{x^2}{25} - \frac{y^2}{36} = - 1, (5\sqrt{3}, -12)

For the relation, determine the equation of the tangent at the given point.

\displaystyle \frac{x^2}{81} - \frac{5y^2}{162} = 1, (-11, -4)

At what point is the tangent to the curve x+ y^2 = 1 parallel to the line x + 2y = 0 ?

The equation 5x^2 - 6xy + 5y^2 = 16 represents an ellipse.

a. Detrmine \frac{dy}{dx} at (1, -1).

b. Determine two points on the ellipse at which the tangent is horizontal.

Determine the slope of the tangent to the ellipse 5x^2 + y^2 = 21 at the point A(-2, -1).

Determine the equation of the normal to the curve x^3 + y^3 - 3xy = 17 at the point (2, 3).

Determine the equation of the normal to the curve y^2 = \frac{x^3}{2- x} at the point (1, -1).

Determine \frac{dy}{dx}

(x + y)^3 = 12x

Determine \frac{dy}{dx}

\sqrt{x + y} - 2x = 1

The equation 4x^2y - 3y = x^3 implicitly defines y as a function of x.

a. Use implicit differentiation to determine \frac{dy}{dx}.

b. Write y as an explicit function of x, and compute \frac{dy}{dx} directly.

Show that \frac{dy}{dx} = \frac{y}{x} for the relation \sqrt{\frac{x}{y}} + \sqrt{\frac{y}{x}} = 10, x , y \neq 0.

Determine the equations of the lines that are tangent to the ellipse \displaystyle x^2 + 4y^2 = 16 and also pass through the point (4, 6).

The angle between two intersecting curves is defined as the angle between their tangents at the point of intersection. If this angle is 90°, the two curves are said to be orthogonal at this point.

Prove that the curves defined by x^2-y^2 = k and xy = p intersect orthogonally for all values of the constants k and p. Illustrate your proof with a sketch.

Let l be any tangent to the curve \sqrt{x} + \sqrt{y} = \sqrt{k}, where k is a constant. Show that the sum of the intercepts of l is k.

Two circles of radius 3\sqrt{2} are tangent to the graph y^2 = 4x at the point (1, 2). Determine the equations of these two circles.