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Chapter 2: Problem 1

Find the derivative of \(y=x^{15 / 16}\). \(\frac{d y}{d x}=\) __________

Short Answer

Expert verified
\[ \frac{d y}{d x} = \frac{15}{16} x^{-\frac{1}{16}} \]

Step by step solution

01

- Identify the Function

The function given is a power function. Identify the function as: \[ y = x^{15/16} \]
02

- Apply the Power Rule for Differentiation

The power rule states that if you have a function of the form \( y = x^n \), its derivative is \( \frac{d y}{d x} = n x^{n-1} \). For our function, \( n = \frac{15}{16} \).
03

- Differentiate the Function

Using the power rule: \[ \frac{d y}{d x} = \frac{15}{16} x^{(\frac{15}{16} - 1)} \] Simplify the exponent: \[ \frac{d y}{d x} = \frac{15}{16} x^{-\frac{1}{16}} \]

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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 an essential concept in calculus that simplifies finding the derivative of power functions. According to this rule, if we have a function of the form \[ y = x^n \], the derivative can be easily calculated. The rule states that the derivative of \[ y = x^n \] is \[ \frac{dy}{dx} = n x^{n-1} \]. This means you multiply by the exponent and then subtract one from the exponent.
For example, if your function is \[ y = x^{15/16} \], applying the Power Rule gives you:
\Opened & Opened \Theta = \frac{15}{16} \ and we subtract 1: \[ \frac{dy}{dx} = \frac{15}{16} x^{(\frac{15}{16} - 1)} \].
Differentiation
Differentiation is a fundamental tool in calculus used to compute the rate of change of a function. It helps us find the slope of a function at any given point. When we differentiate \[ y = x^n \] using the Power Rule, we're performing a specific type of differentiation that focuses on power functions. Differentiation involves applying rules like the power rule, product rule, and chain rule to find the derivatives of different types of functions. In our example with \[ y = x^{15/16} \], we only need the power rule to find its derivative.
Exponent Simplification
Simplifying exponents is a crucial step in differentiation to keep our results manageable and easy to understand. After applying the Power Rule, you often need to simplify the new exponent. For example, starting with \[ y = x^{15/16} \] and differentiating gives us:
\[\frac{dy}{dx} = \frac{15}{16} x^{(\frac{15}{16} - 1)}\].
Simplify the exponent by performing the subtraction: \[\frac{15}{16} - 1 = \frac{15}{16} - \frac{16}{16} = -\frac{1}{16}\].
So, the simplified form of the derivative is:
\[\frac{dy}{dx} = \frac{15}{16} x^{-\frac{1}{16}}\].
Calculus
Calculus is a branch of mathematics that studies change. It includes two main concepts: differentiation and integration. Differentiation is used to find the rate at which things change, while integration is used to find the total amount or area under a curve. The problem we solved is part of the differentiation aspect of calculus.
  • We used the Power Rule, one of the basic rules in calculus, to find the derivative of \[ y = x^{15/16} \].
  • The process involved taking the given power function and applying the power rule to find its derivative, \[ \frac{dy}{dx} \].
  • Next, we simplified the exponent result to get the final derivative form.
Ultimately, calculus helps us understand how functions behave and change, which is invaluable in fields such as physics, engineering, economics, and many more.

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Most popular questions from this chapter

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