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Chapter 2: Problem 13

Find the derivative of each function. \(f(t)=\sin t \sec t\)

Short Answer

Expert verified
The derivative of the given function \(f(t)=\sin t \sec t\) is \(f'(t) = \cos t \sec t + \sin t \sec t \tan t\).

Step by step solution

01

Differentiate \(\sin t\)

The derivative of \(\sin t\) with respect to \(t\) is \(\cos t\).
02

Differentiate \(\sec t\)

The derivative of \(\sec t\) with respect to \(t\) is \(\sec t \tan t\).
03

Apply the product rule

Now, apply the product rule of differentiation: derivative of first function times the second plus the first function times the derivative of the second. So, \(f'(t) = (\cos t * \sec t) + (\sin t * \sec t * \tan t)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Product Rule
The product rule is a crucial differentiation technique useful when finding the derivative of the product of two functions. When you have two differentiable functions, say \( u(t) \) and \( v(t) \), and you're looking at their product \( u(t) \cdot v(t) \), the product rule states that the derivative of this product is \( u'(t) \cdot v(t) + u(t) \cdot v'(t) \). This means:
  • The derivative of the product is not simply the product of the derivatives.
  • Instead, it involves a combination: the derivative of the first function times the second, plus the first function times the derivative of the second.
Applying this concept helps us tackle problems involving derivatives where two or more functions are multiplied together.
In our exercise: \( f(t) = \sin t \cdot \sec t \), the product rule helps us break this complex function into simpler parts, making it easier to differentiate.
Trigonometric Functions
Trigonometric functions like \( \sin t \) and \( \sec t = 1/\cos t \) appear frequently in calculus, especially in problems involving derivatives.
  • The derivative of \( \sin t \) is \( \cos t \). This follows because the rate of change of \( \sin t \) is modeled by the cosine function.
  • The derivative of \( \sec t \) is \( \sec t \cdot \tan t \). This result derives from the differentiation of \( 1/\cos t \), using quotient rule techniques, which is more complex but very valuable.
When dealing with trigonometric functions, it's important to remember their unique properties and derivatives. These derivatives often involve other trigonometric functions that you need to handle carefully in calculations.
Understanding these properties allows us to correctly apply rules of differentiation to functions involving trigonometric expressions.
Differentiation Techniques
Differentiation techniques encompass all the rules and methods employed to find the derivative of different types of functions efficiently. Key techniques include:
  • Basic rules like the power rule and constant rule.
  • Advanced rules such as the product rule, quotient rule, and chain rule.
Each technique is specialized for particular forms of functions or products of functions. In our exercise:
  • We use the product rule because \( f(t) = \sin t \cdot \sec t \) involves the product of \( \sin t \) and \( \sec t \).
  • These are multiplied trigonometric functions, each differentiated separately and combined using the product rule.
Mastering these techniques is essential for efficiently solving calculus problems and understanding the behavior of functions through their rates of change. With practice, choosing the right technique becomes intuitive, making differentiation much simpler.

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