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Chapter 2: Problem 2

Find the derivative of each function. $$f(x)=x^{9}-3 x^{5}+4 x^{2}-4 x$$

Short Answer

Expert verified
The derivative of the function \(f(x) = x^9-3x^5+4x^2-4x\) is \(f'(x)=9x^8-15x^4+8x-4\).

Step by step solution

01

Identify each term in the function

Our function \(f(x)\) comprises of four terms: \(x^9\), \(x^5\), \(x^2\), and \(x\). Each of these terms will be differentiated separately.
02

Apply the power rule to each term

Applying the power rule to each term, we differentiate \(x^n\) to obtain \(nx^{n-1}\). For the first term, \(x^9\), the derivative is \(9x^{9-1}=9x^8\). For the second term, \(3x^5\), we multiply the constant 3 by the derivative of \(x^5\) to get \(3*5x^{5-1}=15x^4\). For the third term, \(4x^2\), we multiply the constant 4 by the derivative of \(x^2\) to get \(4*2x^{2-1}=8x\). Lastly, for the term \(4x\), the derivative is simply the constant in front of the \(x\), which is 4.
03

Combine derivatives of each term

We combine the derivatives from step 2 to construct the derivative of the entire function. This yields the derivative of \(f(x)\) as \(f'(x)=9x^8-15x^4+8x-4\).

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Key Concepts

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

Understanding the Derivative
The derivative of a function is a fundamental concept in calculus. It measures how a function value changes as its input changes. The derivative is like the 'slope' of the function at any given point. In math terms, if you have a function \( f(x) \), its derivative, denoted \( f'(x) \) or \( \frac{df(x)}{dx} \), provides a precise way of describing the rate of change of \( f(x) \) concerning \( x \). This concept helps in understanding how to predict or describe behavior, change, or trends represented by the function over its domain.
Applying the Power Rule
The power rule is a crucial tool in finding derivatives of terms in a polynomial function. It states that if you have a term \( x^n \), its derivative will be \( nx^{n-1} \). This rule simplifies the process tremendously:
  • The exponent \( n \) of the term becomes a multiplier in front of the term.
  • Then, the original exponent \( n \) is reduced by one, resulting in \( n-1 \) as the new exponent.
For example, the term \( x^9 \) becomes \( 9x^8 \) on differentiation. This rule allows you to quickly compute derivatives without complex calculations, making it ideal for polynomial expressions like our original function.
Differentiation Process Explained
Differentiation is the process of finding a derivative. When you differentiate a function, you apply differentiation rules鈥攍ike the power rule鈥攕ystematically to each term of the function. This means:
  • Identify each term in the function. Terms are separated by addition or subtraction.
  • Apply the power rule or any relevant rule to differentiate each term.
  • Sum up all the differentiated terms to get the derivative of the entire function.
For our function \( f(x)=x^9-3x^5+4x^2-4x \), we differentiate each term using the power rule and then combine the results to form \( f'(x)=9x^8-15x^4+8x-4 \).
Breaking Down Function Terms
Functions are often composed of various terms, and each needs to be dealt with individually in differentiation. A term can be as simple as a number, \( 4 \), or as complex as \( 9x^8 \). In our example:
  • \( x^9 \) is a term with a higher exponent, easily handled by the power rule.
  • \( -3x^5 \) and \( 4x^2 \) are terms with coefficients. The coefficient (like \(-3\) and \(4\)) is multiplied by the derivative of the power of \( x \).
  • \( -4x \) is a linear term, simply differentiated to its coefficient \(-4\).
This breakdown ensures you correctly apply calculus principles to each part, finding the derivative accurately.

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Most popular questions from this chapter

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