Question

Explain the differences between computing the derivatives of functions that are defined implicitly and explicitly.

Short answer

Based on the provided solution, explain the differences between computing the derivatives of functions defined implicitly and explicitly. The differences between computing the derivatives of functions defined implicitly and explicitly are: 1. For explicit functions, the derivative is computed by directly applying the rules of differentiation to the y expression, whereas for implicit functions, the derivative requires differentiating both sides of the equation with respect to x and solving for dy/dx. 2. When computing the derivatives of implicit functions, the chain rule is applied to treat y as a function of x during differentiation, which is not necessary for explicit functions. 3. The derivative of an implicit function requires solving for dy/dx after differentiating both sides of the equation, while the derivative of an explicit function is obtained directly using the differentiation rules.

Step-by-step solution

Understanding Implicit and Explicit Functions

An explicit function is a function in which one variable is represented in terms of the other variable. In other words, y is expressed explicitly as a function of x. For example, y=2x+3. An implicit function is a function in which the relationship between the variables is not clearly defined, and y cannot be explicitly given as a function of x. For example, x^2 + y^2 = 1.

Derivative Calculation for Explicit Functions

For explicit functions, the derivative can be found by directly applying the rules of differentiation, such as power rule, product rule, quotient rule, or chain rule, depending on the given function. Example: Given the explicit function: y = 4x^3 - 2x To find the derivative dy/dx, we can use power rule: dy/dx = d/dx(4x^3 - 2x) = 12x^2 - 2

Derivative Calculation for Implicit Functions

For implicit functions, derivatives cannot be found directly by applying the differentiation rules to the y expression (since it is not given explicitly as a function of x). Instead, we differentiate both sides of the equation with respect to x and then solve for dy/dx. The general process for calculating the derivative of an implicit function is: 1. Differentiate both sides of the equation with respect to x. 2. Make use of the chain rule by treating y as a function of x (i.e., mark all instances of y with dy/dx terms). 3. Solve for dy/dx. Example: Given the implicit function: x^2 + y^2 = 1 1. Differentiate both sides with respect to x: d/dx(x^2 + y^2) = d/dx(1) 2. Using the chain rule: 2x + 2y(dy/dx) = 0 3. Solve for dy/dx: dy/dx = -2x / (2y) = -x/y

Summarizing the Differences

In summary, the main differences between computing the derivatives of functions defined implicitly and explicitly are: 1. For explicit functions, the derivative can be found by directly applying the rules of differentiation to the y expression, while for implicit functions, we differentiate both sides of the equation with respect to x and solve for dy/dx. 2. For implicit functions, the chain rule is used to treat y as a function of x when differentiating. 3. The derivative of an implicit function requires solving for dy/dx after differentiating both sides of the equation, whereas the derivative of an explicit function is calculated directly by applying the differentiation rules.