Textbook Question
Evaluate the following limits or state that they do not exist. (Hint: Identify each limit as the derivative of a function at a point.)
lim x→π/4 cot x−1 / x−π/4
3. Techniques of Differentiation
Derivatives of Trig Functions
3:00 minutesProblem 3.5.26
Textbook Question
Derivatives
In Exercises 23–26, find dr/dθ.
r = (1 + sec θ) sin θ
Verified step by step guidance1
First, identify the function r in terms of θ: r(θ) = (1 + sec(θ)) * sin(θ).
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).
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).
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).
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)].
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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.
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