13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.
cos y = x; (0, π/2)

Verified step by step guidance

1. Start by differentiating both sides of the equation with respect to x. For the left side, differentiate cos(y) using the chain rule: the derivative of cos(y) is -sin(y) times the derivative of y with respect to x, which is dy/dx. For the right side, the derivative of x is 1.
2. Write the differentiated equation: -sin(y) * (dy/dx) = 1.
3. Solve for dy/dx by isolating it on one side of the equation. This involves dividing both sides by -sin(y), giving dy/dx = -1/sin(y).
4. Substitute the given point (0, π/2) into the equation to find the slope at that point. Since y = π/2, substitute π/2 for y in the expression for dy/dx.
5. Evaluate sin(π/2) to find the slope. Since sin(π/2) = 1, substitute this value into the expression for dy/dx to find the slope at the point (0, π/2).

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not explicitly separated. Instead of solving for one variable in terms of the other, we differentiate both sides of the equation with respect to the independent variable, applying the chain rule as necessary. This method is particularly useful for curves defined by equations that cannot be easily rearranged.

Slope of a Curve

The slope of a curve at a given point represents the rate of change of the dependent variable with respect to the independent variable at that specific point. Mathematically, it is found by evaluating the derivative of the function at that point. In the context of implicit differentiation, the slope can be expressed as dy/dx, which indicates how y changes with respect to x.

Evaluating Derivatives at a Point

To find the slope of a curve at a specific point, we first compute the derivative of the function and then substitute the coordinates of the point into this derivative. This process allows us to determine the instantaneous rate of change at that point. For the given problem, substituting the point (0, π/2) into the derived expression for dy/dx will yield the slope of the curve at that location.