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Chapter 3: Problem 10

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

Short Answer

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

Step by step solution

01

Rewrite the Function

Recognize that the function is a reciprocal power of x. Given that \(f(x) = \frac{1}{x^4}\), rewrite it using negative exponents: \(f(x) = x^{-4}\). This form is more convenient for differentiation.
02

Differentiate using Power Rule

Use the power rule for differentiation, which states that the derivative of \(x^n\) is \(nx^{n-1}\). Here, \(n = -4\). Apply the power rule: the derivative \(f'(x) = -4x^{-4-1} = -4x^{-5}\).
03

Simplify the Derivative

Express the derivative with positive exponents for a nicer presentation. Therefore, \(f'(x) = -\frac{4}{x^5}\).

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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 cornerstone in calculus when it comes to finding the derivative of polynomial expressions. It's wonderfully simple and powerful, allowing us to differentiate functions of the form \(x^n\) quickly.
  • According to the power rule, the derivative of \(x^n\) is \(nx^{n-1}\).
  • This rule only applies to functions where the power \(n\) is a constant.

Harnessing the power rule is straightforward. You take the exponent, multiply it by the coefficient in front of \(x\), and then reduce the exponent by one. For instance, if \(f(x) = x^3\), then \(f'(x) = 3x^{2}\). This simple manipulation gives us a new function that represents the rate of change of the original function.
Negative Exponents
Negative exponents are a useful way of expressing reciprocal powers and can simplify differentiation.
  • A negative exponent like \(x^{-n}\) is equivalent to \(\frac{1}{x^n}\).
  • Using negative exponents makes it easier to apply differentiation techniques, such as the power rule.

In our exercise, the function \(\frac{1}{x^4}\) transforms to \(x^{-4}\) by applying the concept of negative exponents. This transformation is crucial as it allows us to apply the power rule directly. By doing this, complex fraction derivative problems become manageable and simple to solve.
Differentiation Techniques
Differentiation bridges the gap between algebraic expressions and calculus concepts, offering tools to calculate rates of change.
  • Common techniques include the power rule, product rule, quotient rule, and chain rule.
  • For single-variable polynomials or terms like \(x^n\), the power rule is the most direct approach.

When tackling more complex expressions, combinations of these rules may be required. It's essential to restate the function in a differentiable form first. In our solution, rewriting \(\frac{1}{x^4}\) as \(x^{-4}\) exemplifies a differentiation technique鈥攃onverting a division form to a power form so the power rule can be applied. Simplifying after differentiation is also key, ensuring your answer is tidy and as straightforward as possible, like expressing derivatives with positive exponents.

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Most popular questions from this chapter

In \(2009,\) the population of Hungary \(^{4}\) was approximated by $$P=9.906(0.997)^{t}$$ where \(P\) is in millions and \(t\) is in years since \(2009 .\) Assume the trend continues. (a) What does this model predict for the population of Hungary in the year \(2020 ?\) (b) How fast (in people/year) does this model predict Hungary's population will be decreasing in 2020 ?

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