Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not separated. It allows us to find the derivative of y with respect to x when y is defined implicitly as a function of x.

The derivative of a function at a point is the slope of the tangent line to the graph of the function at that point. It represents the rate of change of the function with respect to its variable.

Differentiate both sides with respect to x, treating y as a function of x. This gives $$6x - 18y^2 \frac{dy}{dx} = 0$$ . Rearranging yields $$\frac{dy}{dx} = \frac{6x}{18y^2} = \frac{x}{3y^2}$$ .

The formula is $$\frac{dy}{dx} = -\frac{F_x}{F_y}$$ , where $$F_x$$ is the partial derivative of the function with respect to x, and $$F_y$$ is the partial derivative with respect to y.

Differentiating gives $$3x^2 + 3y^2 \frac{dy}{dx} = 0$$ . Solving for $$\frac{dy}{dx}$$ yields $$\frac{dy}{dx} = -\frac{x^2}{y^2}$$ .

Rearranging gives $$9y - 4y + x = 0$$ . Differentiating gives $$5 \frac{dy}{dx} + 1 = 0$$ , leading to $$\frac{dy}{dx} = -\frac{1}{5}$$ .

Differentiate both sides: $$-8x + 12 + 12y \frac{dy}{dx} - 10 \frac{dy}{dx} = 0$$ . Rearranging gives $$\frac{dy}{dx} = \frac{8x - 12}{12y - 10}$$ .

The slope of the tangent line at a point on a curve is given by the derivative of the function at that point.

A secant line is a line that intersects a curve at two or more points. The slope of the secant line between two points on the curve approximates the average rate of change of the function between those points.

The derivative at a point gives the slope of the tangent line to the curve at that point, indicating how steep the curve is at that location.