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Chapter 10: Problem 9

In Exercises 9 through 24 , find the derivative of the given function. \(f(x)=3 \sin 2 x\)

Short Answer

Expert verified
The derivative of \( f(x) = 3 \sin(2x) \) is \( f'(x) = 6 \cos(2x) \).

Step by step solution

01

Understand the Function

The given function is a combination of a constant, a sine function, and an inside function. Specifically, the function is of the form: \[ f(x) = 3 sin(2x) \]
02

Apply the Chain Rule

To differentiate this function, we need to apply the chain rule and the derivative of the sine function, which is: \[ \frac{d}{dx} [\sin(u)] = \cos(u) \cdot \frac{du}{dx} \]Here, \( u = 2x \), so need to find the derivative of \(2x\) and multiply accordingly.
03

Differentiate Inside Function

First, we differentiate the inside function: \[ \frac{d}{dx} [2x] = 2 \]
04

Differentiate the Outer Function

Next, differentiate the outer function (the sine function). According to the chain rule, we multiply the derivative of the outer function by the derivative of the inner function: \[ \frac{d}{dx} [\sin(2x)] = \cos(2x) \cdot 2 \]
05

Combine with Constant

Lastly, we include the constant coefficient from the original function and multiply it by our result in Step 4: \[ f'(x) = 3 \cdot (\cos(2x) \cdot 2) = 6 \cos(2x) \]

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Key Concepts

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

chain rule
When dealing with the derivative of composite functions, the chain rule is indispensable. The chain rule states that to differentiate a composite function, we must first differentiate the outer function, and then multiply by the derivative of the inner function. Formally, if you have a function of the form \(f(g(x))\), its derivative is found using the formula: \( (f(g(x)))' = f'(g(x)) \times g'(x) \). This rule helps us break down complex functions into simpler parts, making it easier to manage and solve. Always start with the innermost function and work your way out.
trigonometric derivatives
Trigonometric functions have specific derivatives that you should memorize. These derivatives are very useful and often appear in calculus problems.\Below are some of the key derivatives you need to remember:
  • \frac{d}{dx} [sin(x)] = cos(x) \
  • \frac{d}{dx} [cos(x)] = -sin(x) \
  • \frac{d}{dx} [tan(x)] = sec^2(x) \
In our given problem, we are particularly interested in the derivative of the sine function. According to the rules, \frac{d}{dx} [sin(u)] = cos(u) \. Notice how we treat \(u\) as a placeholder for any function, which is essential for applying the chain rule.
constant multiple rule
The constant multiple rule simplifies the process of taking derivatives when a function is multiplied by a constant. According to this rule, if you have a constant \(c\) and a function \(f(x)\), the derivative of their product is: \frac{d}{dx} [c \times f(x)] = c \times \frac{d}{dx} [f(x)] \.This rule is especially useful because it allows us to focus solely on differentiating the function part, and then simply multiply by the constant afterward. In our exercise, the constant is 3. After applying the chain rule and finding the derivative \(2 \times cos(2x)\), we multiply by this constant, leading to the final result: \6 \times cos(2x)\.

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