Differentiating Implicitly


Use implicit differentiation to find dy/dx in Exercises 1–14.


x = sec y

Verified step by step guidance

1. Start by understanding implicit differentiation. This technique is used when you have an equation involving both x and y, and you want to find dy/dx without solving for y explicitly.
2. Given the equation x = sec(y), differentiate both sides with respect to x. Remember that when differentiating y with respect to x, you need to multiply by dy/dx due to the chain rule.
3. Differentiate the left side: The derivative of x with respect to x is 1.
4. Differentiate the right side: The derivative of sec(y) with respect to y is sec(y)tan(y). Since y is a function of x, apply the chain rule and multiply by dy/dx.
5. Set the derivatives equal to each other: 1 = sec(y)tan(y) * dy/dx. Solve for dy/dx by isolating it on one side of the equation.

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 equations that define y implicitly in terms of x.

Chain Rule

The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. It states that if a function y is defined as a function of u, which in turn is a function of x, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This rule is essential when differentiating implicitly, as it helps manage the relationships between variables.

Secant Function

The secant function, denoted as sec(y), is the reciprocal of the cosine function, defined as sec(y) = 1/cos(y). In the context of the given equation x = sec(y), it relates the angle y to the horizontal distance x in a right triangle. Understanding the properties of the secant function is crucial for implicit differentiation, as it influences how we differentiate the equation with respect to x.