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Chapter 4: Problem 12

find \(\frac{d}{d x} f(x)\) $$ f(x)=x^{2 / 5} $$

Short Answer

Expert verified
The derivative is \( \frac{2}{5}x^{-3/5} \).

Step by step solution

01

Identify the Function and Its Form

The function given is \( f(x) = x^{2/5} \). This is a power function where the base is \( x \) and the exponent is \( \frac{2}{5} \).
02

Recall the Power Rule for Derivatives

The power rule for differentiation states that if \( f(x) = x^n \), then the derivative \( \frac{d}{dx}f(x) = nx^{n-1} \). This means you need to multiply the variable \( x \) by the exponent and then subtract one from the exponent.
03

Apply the Power Rule

Applying the power rule to \( f(x) = x^{2/5} \), you multiply \( x \) by the exponent \( \frac{2}{5} \). Then, subtract 1 from the exponent, which gives the new exponent as \( \frac{2}{5} - 1 = -\frac{3}{5} \).
04

Calculate the Derivative

The derivative is calculated as follows:\[ \frac{d}{dx}f(x) = \frac{2}{5}x^{-3/5} \]This means the derivative of \( f(x) = x^{2/5} \) is \( \frac{2}{5}x^{-3/5} \).

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Key Concepts

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

Power Rule for Derivatives
The power rule for derivatives is a fundamental concept in calculus that simplifies finding the derivative of power functions. Power functions are functions where a variable, typically noted as \( x \), is raised to a constant exponent. The power rule states that if you have a function \( f(x) = x^n \), its derivative, denoted as \( \frac{d}{dx}f(x) \), is \( nx^{n-1} \). This means:
  • Multiply the variable's exponent by the coefficient (often implicit as 1 when it's just \( x^n \)).
  • Subtract one from the exponent.
Applying the power rule helps quickly find the rate of change of many mathematical functions. In essence, it's a handy shortcut in calculus for differentiating power expressions, making calculations much more efficient.
Differentiation Methods
Differentiation is the process of finding a derivative, which measures how a function changes as its input changes. One popular method is the power rule, as discussed, but there are other techniques as well, including:
  • Product Rule: Used when differentiating products of two functions. If \( u(x) \) and \( v(x) \) are functions, the derivative is \((uw)' = u'v + uv'\).
  • Quotient Rule: Useful for functions that are ratios of two functions. If your function is \( f(x) = \frac{u(x)}{v(x)} \), the derivative is \( \, \frac{u'v - uv'}{v^2} \, \).
  • Chain Rule: Perfect for composite functions, where one function is nested inside another. If \( y = g(f(x)) \), then the derivative is \( \frac{dy}{dx} = g'(f(x)) \cdot f'(x) \).
Each differentiation method has its particular use case, and choosing the correct one depends on the type of function you're working with. Being familiar with these methods makes complex problems simpler to solve.
Mathematical Functions
Mathematical functions are essential building blocks in calculus, and understanding them is critical for differentiating functions accurately. Functions can take many forms, such as:
  • Polynomial Functions: These are sums of constants times variables raised to non-negative integer powers, like \( f(x) = 3x^3 + 2x^2 + x - 5 \).
  • Rational Functions: These are ratios of polynomial functions, for example, \( g(x) = \frac{2x^2 + 3}{x - 1} \).
  • Exponential Functions: These involve the base of a constant raised to the power of a variable, like \( h(x) = 2^x \).
  • Logarithmic Functions: These functions are the inverse of exponential functions, such as \( \, \log_b(x) \, \).
Understanding the form of a function helps in applying the correct differentiation method. Functions may also be combined or transformed, and recognizing these combinations aids in more effective and accurate derivatives calculation.

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Most popular questions from this chapter

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