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Chapter 3: Problem 37

Derivatives of functions with rational exponents Find \(\frac{d y}{d x}\). $$y=x^{5 / 4}$$

Short Answer

Expert verified
Short answer: To find the derivative of the function \(y = x^{5/4}\), we apply the Power Rule for differentiation and simplify the expression. The derivative, \(\frac{dy}{dx}\), is \(\frac{5}{4} x^{\frac{1}{4}}\).

Step by step solution

01

Identify the exponent and the base

For this function, the base is \(x\) and the exponent is \(5/4\).
02

Apply the Power Rule for differentiation

The Power Rule states that if \(y = x^n\), then \(\frac{dy}{dx} = n x^{n-1}\). Applying this rule to our function, we have: $$ \frac{dy}{dx} = \frac{5}{4} x^{\frac{5}{4} - 1} $$
03

Simplify the exponent

Now, we need to simplify the exponent by subtracting 1 from the current exponent: $$ \frac{5}{4} - 1 = \frac{5}{4} - \frac{4}{4} = \frac{1}{4} $$
04

Write the final answer

Substitute the simplified exponent back into the derivative expression: $$ \frac{dy}{dx} = \frac{5}{4} x^{\frac{1}{4}} $$ The derivative of the function \(y = x^{5/4}\) is: $$ \frac{dy}{dx} = \frac{5}{4} x^{\frac{1}{4}} $$

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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, making it easier to find the derivative of functions with exponents. It applies to functions of the form \( y = x^n \), where \( n \) can be any real number, including fractions, which are known as rational exponents.

The rule states that the derivative of \( x^n \) is \( n \times x^{n-1} \). This means you multiply the entire function by the exponent and then decrease the exponent by one. It's a quick way to handle polynomial terms in differentiation.
  • Identify the exponent \( n \).
  • Multiply the function by \( n \).
  • Subtract one from the original exponent.
Knowing the Power Rule simplifies differentiating complex equations with polynomial elements, saving both time and effort.
Derivatives
Derivatives represent the rate at which a function changes. In simple terms, a derivative can be seen as the function's 'instantaneous speed.'

Whenever you observe how one quantity changes concerning another, you are dealing with derivatives. They play a crucial role in many field applications, from physics to economics.
  • Derivatives are used to find slopes of tangents.
  • They help in understanding and predicting the behavior of systems.
  • Essential for optimization problems, calculating maxima and minima.
In the context of rational exponents, derivatives help determine how quickly the function rises or falls for non-integer powers of \( x \). This is particularly handy in examining the nature of curves and graphs.
Rational Exponents
Rational exponents are exponents that are fractions. Rather than dealing with whole numbers, the exponents come in the form \( a/b \), where both \( a \) and \( b \) are integers.

For example, \( x^{5/4} \) is a rational exponent, meaning the base \( x \) is raised to a fractional power. Rational exponents can be thought of in terms of roots:
  • If \( x^{1/2} \) is equivalent to \( \sqrt{x} \).
  • Similarly, \( x^{5/4} \) can be rewritten as \( \sqrt[4]{x^5} \).
These exponents make it easier to express roots and powers within the same framework, offering a versatile way to deal with various equations in calculus. Using rational exponents can simplify many differentiation problems.

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Most popular questions from this chapter

Find the derivative of the inverse of the following functions at the specified point on the graph of the inverse function. You do not need to find \(f^{-1}\) $$f(x)=x^{2}-2 x-3, \text { for } x \leq 1 ;(12,-3)$$

Use a trigonometric identity to show that the derivatives of the inverse cotangent and inverse cosecant differ from the derivatives of the inverse tangent and inverse secant, respectively, by a multiplicative factor of -1

Find \(f^{\prime}(x), f^{\prime \prime}(x),\) and \(f^{\prime \prime \prime}(x)\) \(f(x)=\frac{x^{2}-7 x}{x+1}\)

The bottom of a large theater screen is \(3 \mathrm{ft}\) above your eye level and the top of the screen is \(10 \mathrm{ft}\) above your eye level. Assume you walk away from the screen (perpendicular to the screen) at a rate of \(3 \mathrm{ft} / \mathrm{s}\) while looking at the screen. What is the rate of change of the viewing angle \(\theta\) when you are \(30 \mathrm{ft}\) from the wall on which the screen hangs, assuming the floor is horizontal (see figure)?

Derivatives and inverse functions Find the slope of the curve \(y=f^{-1}(x)\) at (4,7) if the slope of the curve \(y=f(x)\) at (7,4) is \(\frac{2}{3}\)
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