Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 5: Problem 5
Find the derivative and state a corresponding integration formula. $$ \frac{d}{d x}[\sqrt{x^{3}+5}] $$
Short Answer
Expert verified
The derivative is \( \frac{3x^2}{2\sqrt{x^3+5}} \) and the corresponding integration formula is \( \int \frac{3x^2}{2\sqrt{x^3+5}} \, dx = \sqrt{x^3+5} + C \).
Step by step solution
01
Identify the Function
The function we need to differentiate is \( f(x) = \sqrt{x^3 + 5} \). This can also be written as \( (x^3 + 5)^{1/2} \). It is a composition of functions: an outer function (the square root) and an inner function \( x^3 + 5 \).
02
Apply the Chain Rule
To differentiate \( f(x) = (x^3 + 5)^{1/2} \), we use the chain rule. The chain rule states that the derivative \( \frac{d}{dx}[u(v)] = u'(v) \cdot v'(x) \). Here, \( u(v) = v^{1/2} \) and \( v = x^3 + 5 \), so \( u'(v) = \frac{1}{2}v^{-1/2} \) and \( v'(x) = 3x^2 \).
03
Differentiate the Outer Function
First, differentiate the outer function \( u(v) = v^{1/2} \) with respect to \( v \), which gives: \( u'(v) = \frac{1}{2} v^{-1/2} \).
04
Differentiate the Inner Function
Next, differentiate the inner function \( v(x) = x^3 + 5 \) with respect to \( x \), which results in: \( v'(x) = 3x^2 \).
05
Combine Using the Chain Rule
Combine the derivatives from the previous steps using the chain rule: \( \frac{d}{dx}[(x^3+5)^{1/2}] = \frac{1}{2}(x^3+5)^{-1/2} \cdot 3x^2 = \frac{3x^2}{2\sqrt{x^3+5}} \).
06
State a Corresponding Integration Formula
The derivative we found is \( \frac{3x^2}{2\sqrt{x^3+5}} \). To find a corresponding integration formula, recognize that differentiating \( \sqrt{x^3+5} \) gave us this result. Thus, if you integrate \( \frac{3x^2}{2\sqrt{x^3+5}} \) with respect to \( x \), you would obtain \( \sqrt{x^3+5} + C \) where \( C \) is the constant of integration.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Chain Rule
The chain rule is a fundamental principle in calculus used for differentiating composite functions. A composite function is essentially a function made up of other functions, just like the one in our exercise: \( f(x) = (x^3 + 5)^{1/2} \). Understanding the chain rule allows us to break down complex functions into simpler parts for easier differentiation.
- Think of the chain rule like peeling an onion layer by layer. You start from the outside and work your way inward.
- The rule itself states that to differentiate a composite function \( f(g(x)) \), you multiply the derivative of the outer function \( f \) by the derivative of the inner function \( g \). Mathematically, it's expressed as \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \).
- In our problem, the outer function is the square root, expressed as \( (v)^{1/2} \), and the inner function is \( x^3 + 5 \).
Function Composition
Function composition is the process of combining two functions to form a new one. It is represented as \( (f \circ g)(x) = f(g(x)) \), meaning you apply one function to the results of another.
- In our exercise, the function \( f(x) = \sqrt{x^3 + 5} \) is composed of an outer square root function and an inner polynomial function \( x^3 + 5 \).
- This creates a connection between the functions, where the output of the inner function (\( x^3 + 5 \)) is used as the input for the outer function (the square root).
Integration Formula
Once you've differentiated a function, it's often useful to find the corresponding integration formula.
- Integration essentially reverses differentiation. If you know the derivative, you can obtain the original function through integration, plus a constant term \( C \), known as the constant of integration.
- In our problem, after finding the derivative \( \frac{3x^2}{2\sqrt{x^3+5}} \), we state the corresponding integration formula as \( \int \frac{3x^2}{2\sqrt{x^3+5}} \, dx = \sqrt{x^3 + 5} + C \).
- This relationship showcases the beautiful connection between differentiation and integration, often referred to as the "Fundamental Theorem of Calculus".
