Derivatives


In Exercises 23–26, find dr/dθ.


r = (1 + sec θ) sin θ

Verified step by step guidance

1. First, identify the function r in terms of θ: r(θ) = (1 + sec(θ)) * sin(θ).
2. To find the derivative dr/dθ, apply the product rule, which states that if you have a function u(θ) * v(θ), then the derivative is u'(θ) * v(θ) + u(θ) * v'(θ).
3. Let u(θ) = 1 + sec(θ) and v(θ) = sin(θ). First, find the derivative of u(θ): u'(θ) = d/dθ [1 + sec(θ)]. The derivative of sec(θ) is sec(θ)tan(θ), so u'(θ) = sec(θ)tan(θ).
4. Next, find the derivative of v(θ): v'(θ) = d/dθ [sin(θ)]. The derivative of sin(θ) is cos(θ), so v'(θ) = cos(θ).
5. Now, apply the product rule: dr/dθ = u'(θ) * v(θ) + u(θ) * v'(θ) = [sec(θ)tan(θ) * sin(θ)] + [(1 + sec(θ)) * cos(θ)].

Key Concepts

Derivatives

A derivative represents the rate of change of a function with respect to a variable. In calculus, it is a fundamental concept that allows us to determine how a function behaves as its input changes. The derivative can be interpreted as the slope of the tangent line to the curve of the function at a given point.

Parametric Equations

In this context, the equation r = (1 + sec θ) sin θ is a parametric equation where r is expressed in terms of the parameter θ. Understanding parametric equations is crucial for finding derivatives with respect to a parameter, as it involves differentiating with respect to that parameter rather than the independent variable directly.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. When finding dr/dθ, the chain rule allows us to differentiate r as a function of θ, especially when r is defined in terms of other trigonometric functions. This rule is essential for correctly applying derivatives in parametric contexts.