Differentiation Techniques and Applications

Implicit Differentiation

Implicit differentiation is used when a function is not given explicitly as y = f(x), but rather as an equation involving both x and y. This technique allows us to find the derivative of y with respect to x even when y is not isolated.

• Key Point: Differentiate both sides of the equation with respect to x, treating y as a function of x (i.e., apply the chain rule to terms involving y).

• Key Point: Solve for \frac{dy}{dx} to find the slope of the curve at a given point.

• Example: For the curve x^2 + y^2 = 25, differentiate both sides to get , then solve for .

Equation of the Tangent Line

The tangent line to a curve at a given point has a slope equal to the derivative at that point. The equation of the tangent line can be found using the point-slope form.

• Key Point: The point-slope form is , where m is the slope at (x_1, y_1).

• Example: If the slope at (3, 4) is 2, the tangent line is .

Logarithmic Differentiation

Logarithmic differentiation is a method used to differentiate functions that are products, quotients, or powers, especially when the function is complicated.

• Key Point: Take the natural logarithm of both sides, then differentiate using properties of logarithms.

• Key Point: Useful for functions like or products/quotients of several functions.

• Example: For , take , then differentiate both sides.

Applications of Differentiation

Absolute Maximum and Minimum Points

Absolute extrema are the highest and lowest values of a function on a given interval. These are found by evaluating the function at critical points and endpoints.

• Key Point: Critical points occur where or is undefined.

• Key Point: Evaluate at all critical points and endpoints to determine absolute maximum and minimum.

• Example: For on [0,2], check , , and any critical points in (0,2).

Linearization and Approximation

Linearization uses the tangent line at a point to approximate the value of a function near that point. This is useful for estimating values without a calculator.

• Key Point: The linearization at is .

• Example: To approximate , use at .

Business Applications of Calculus

Elasticity of Demand

Elasticity measures how the quantity demanded responds to changes in price. It is a key concept in economics for understanding consumer behavior.

• Key Point: The elasticity of demand is , where p is price and q is quantity demanded.

• Key Point: If , demand is elastic; if , demand is inelastic; if , demand is unit elastic.

• Key Point: Maximum revenue occurs when elasticity is unitary ().

• Example: Given , find and determine the price for maximum revenue.

Revenue and Marginal Revenue

Revenue is the total income from sales, while marginal revenue is the rate of change of revenue with respect to the number of units sold.

• Key Point: Total revenue: , where is the price function and is the number of units sold.

• Key Point: Marginal revenue: .

• Example: If , then .

Profit Maximization

Profit is the difference between total revenue and total cost. Maximizing profit involves finding the production level where this difference is greatest.

• Key Point: Profit function: , where is the total cost function.

• Key Point: Maximum profit occurs where and (second derivative test).

• Example: If and , then .

Summary Table: Types of Elasticity