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Chapter 8: Problem 3

Differentiate (with respect to \(t\) or \(x\) ): $$y=4 \sin t$$

Short Answer

Expert verified
The derivative of \( y = 4 \text{sin} \, t \) with respect to \( t \) is \( 4 \text{cos} \, t \).

Step by step solution

01

- Identify the function and the variable of differentiation

The function given is \( y = 4 \, \text{sin} \, t \). The variable with respect to which we need to differentiate is \( t \).
02

- Apply the differentiation rule

Recall the differentiation rule for the sine function: \( \frac{d}{dt} \text{sin} \, t = \text{cos} \, t \). Also, note that if a constant multiplies the function, it remains as a constant multiplier.
03

- Differentiate the function

Using the differentiation rule from Step 2, the derivative of \( 4 \, \text{sin} \, t \) with respect to \( t \) is: \[ y' = 4 \frac{d}{dt} \text{sin} \, t = 4 \, \text{cos} \, t \]

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Key Concepts

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

Derivatives Explained
In calculus, a derivative represents how a function changes as its input changes. Basically, it measures the rate at which something is changing. For example, if you have a position function describing how far a car travels over time, the derivative of this function gives you the car's speed at any moment.
To find a derivative, we have certain rules and formulas. One common rule is the power rule, but for functions like sine, we use trigonometric rules.
With the given problem, we focused on differentiating the sine function. Keep in mind, the rules for different functions can vary. For sine and cosine functions, we often rely on specific trigonometric differentiation rules.
Understanding the Sine Function
The sine function, denoted as \text{sin} t\, is a basic trigonometric function. It's crucial in calculus and various scientific fields. The sine function takes an angle (in radians) as input and outputs the y-coordinate of the point on the unit circle.
Some key properties of the sine function include:
  • The sine function oscillates between -1 and 1, never exceeding these values.
  • It has a period of \(2\pi\), meaning it repeats its values every \(2\pi\rightarrow stated: \theta \text{ radians}\).
  • Its graph is a smooth, wave-like shape.
When differentiating, we use the rule: \( \frac{d}{dt} \text{sin} \, t = \text{cos} \, t\). This means the derivative of the sine function is the cosine function, another trigonometric function.
Basics of Calculus
Calculus is a branch of mathematics that deals with change. It's divided into two main areas: differential and integral calculus. Differential calculus focuses on derivatives, which measure how things change. Integral calculus, on the other hand, is about accumulation and finding areas under curves.
Key concepts in differential calculus include:
  • Limits: The value that a function approaches as the input approaches some value.
  • Derivatives: Represent the rates of change.
Knowing derivatives and how to find them is crucial for understanding change and motion.
Overall, mastering calculus requires practice, and understanding the rules of differentiation is a good starting point. The given exercise is an example of how these rules are applied.

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Most popular questions from this chapter

Differentiate (with respect to \(t\) or \(x\) ): $$y=\sqrt[3]{\sin \pi t}$$

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Find the area under the curve \(y=\sin 2 t\) from \(t=0\) to \(t=\frac{\pi}{4}.\)

Differentiate (with respect to \(t\) or \(x\) ): $$f(x)=3 \tan (\pi-x)$$

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