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Chapter 3: Problem 9

Find the derivative. Assume \(a, b, c, k\) are constants. $$f(x)=\frac{1}{x^{4}}$$

Short Answer

Expert verified
The derivative is \( f'(x) = \frac{-4}{x^5} \).

Step by step solution

01

Identify the Function Form

The given function is \( f(x) = \frac{1}{x^4} \). This is a rational function which can be rewritten as a power of \( x \): \( f(x) = x^{-4} \).
02

Apply the Power Rule

The power rule for differentiation states that if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \). For the function \( f(x) = x^{-4} \), apply the power rule: \( f'(x) = -4x^{-5} \).
03

Simplify the Derivative

The expression \( -4x^{-5} \) can be simplified by rewriting it as a fraction: \( f'(x) = \frac{-4}{x^5} \). 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 a fundamental tool in calculus for finding derivatives easily. It is especially handy for functions expressed as a power of the variable. The rule states that for any function of the form \( f(x) = x^n \), the derivative is \( f'(x) = nx^{n-1} \). This means you bring the exponent down in front as a coefficient and subtract one from the exponent.
If you have a function like \( x^3 \), the derivative becomes \( 3x^2 \). The process is straightforward:
  • Identify the exponent, \( n \).
  • Bring down \( n \) as the coefficient, multiplying it by any coefficient of the term.
  • Subtract one from the exponent \( n-1 \).
This rule simplifies differentiation for polynomial functions, turning potentially complex problems into manageable calculations.
Rational Function
A rational function is any function that can be represented as the ratio of two polynomials. An example is \( f(x) = \frac{1}{x^4} \), where the function includes a polynomial in the denominator. Rational functions can be cumbersome to differentiate directly with their fractional form.
To apply calculus to a rational function, it is often helpful to rewrite it as a power of \( x \).
For instance, \( f(x) = \frac{1}{x^4} \) can be rewritten as \( x^{-4} \). This transformation allows us to use the power rule efficiently.
  • Recognize that the denominator can create negative exponents in the function's rewritten form.
  • This enables the application of standard differentiation techniques such as the power rule.
In summary, rewriting rational functions can streamline complicated differentiation processes.
Calculus Basics
Calculus is the mathematical study of change, focusing on variables and their rates of change. It consists of two main branches: differential calculus and integral calculus. Differential calculus deals with finding derivatives, helping to understand how a function changes at any given point.
The derivative of a function provides insight into the slope or rate of change of the function at an exact moment. This concept is crucial in various scientific and engineering applications.
  • Helps in determining rates of change.
  • Essential for understanding motion, growth, and decay.
  • Used to optimize values and find maximum or minimum points.
Understanding these basics is necessary to grasp more advanced calculus concepts and apply them effectively in problem-solving situations.

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