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Chapter 3: Problem 1

Differentiate. \(f(x)=3 x^{2}-2 \cos x\)

Short Answer

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

Step by step solution

01

Differentiate the Power Function

The function given is \[ f(x) = 3x^2 - 2\cos x. \]The first term is a power function, \(3x^2\). Differentiate it using the power rule, which states that the derivative of \(x^n\) is \(nx^{n-1}\). Therefore, the derivative of \(3x^2\) is \(2 \times 3x^{2-1} = 6x\).
02

Differentiate the Cosine Function

The second term in the function is \(-2\cos x\). To differentiate this, use the derivative of \(\cos x\), which is \(-\sin x\). Using the constant multiple rule, the derivative of \(-2\cos x\) is \(-2 \times (-\sin x) = 2\sin x\).
03

Combine the Derivatives

Now that we have both derivatives, combine them to find the derivative of the entire function. \[ f'(x) = 6x + 2\sin x. \] This is the derivative of the given function.

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Key Concepts

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

Power Rule
The power rule is an essential concept in differentiation, simplifying the process of finding derivatives for polynomial functions. When you see a term in the form of \(x^n\), where \(n\) is a constant, the power rule tells us how to find its derivative. The rule is straightforward:
  • Take the exponent \(n\).
  • Multiply it by the coefficient of \(x\).
  • Reduce the exponent by one.
For example, the derivative of \(3x^2\) would be found by setting \(n = 2\) and the existing coefficient as 3. According to the power rule, the derivative is \(2 \times 3x^{2-1}\), simplifying to \(6x\).
This simple rule significantly reduces the time and effort needed to differentiate polynomial expressions, making it a vital tool for calculus students.
Derivative of Cosine
The derivative of cosine is a basic but crucial element in calculus, especially when dealing with trigonometric functions. It's important to remember:
  • The derivative of \(\cos x\) is \(-\sin x\).
Understanding this derivative allows you to tackle problems that involve cosine functions.
In our exercise example, you are asked to differentiate \(-2\cos x\). First, recognize that the derivative of \(\cos x\) is \(-\sin x\). Applying this, \(\frac{d}{dx} [-2\cos x]\) becomes \(-2 \times (-\sin x) = 2\sin x\). This derivative illustrates how trigonometric rules apply to functions like cosine, making the integration of calculus and trigonometry evident.
Constant Multiple Rule
The constant multiple rule complements other differentiation rules by handling constants that multiply functions. Essentially, this rule states:
  • If you multiply a function \(f(x)\) by a constant \(c\), its derivative is simply \(c\) times the derivative of \(f(x)\): \(c \cdot f'(x)\).
Within the context of our exercise, consider \(-2\cos x\). You treat \(-2\) as a constant and \(\cos x\) as the function. The constant multiple rule indicates that after finding the derivative of \(\cos x\) (which is \(-\sin x\)), you multiply it by \(-2\). Hence, the derivative becomes \(-2 \times (-\sin x) = 2\sin x\).
This principle helps streamline differentiation when constants are involved, making it an integral part of understanding how to manage various mathematical expressions properly.

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Most popular questions from this chapter

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