Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 2: Problem 27
Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. $$f(x)=x^{4}$$
Short Answer
Expert verified
The derivative is \(4x^3\) with both original function and its derivative having the domain \((-\infty, \infty)\).
Step by step solution
01
Function and Derivative Definition
The function given is \( f(x) = x^4 \). The definition of the derivative of a function \( f \) at a point \( x \) is given by \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}. \]
02
Substitute into the Derivative Definition
Substitute \( f(x) = x^4 \) into the definition of the derivative: \[ f'(x) = \lim_{h \to 0} \frac{(x+h)^4 - x^4}{h}. \]
03
Expand \((x+h)^4\)
Expand the expression \((x+h)^4\) using the binomial theorem: \[ (x+h)^4 = x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4. \]
04
Simplify the Difference Quotient
Replace \((x+h)^4\) in the difference quotient and simplify: \[ f'(x) = \lim_{h \to 0} \frac{x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4 - x^4}{h}. \]This simplifies to \[ \frac{4x^3h + 6x^2h^2 + 4xh^3 + h^4}{h}. \]
05
Factor Out \(h\)
Factor \( h \) out of each term in the numerator:\[ f'(x) = \lim_{h \to 0} \left(4x^3 + 6x^2h + 4xh^2 + h^3\right). \]
06
Calculate the Limit as \(h \to 0\)
Now take the limit as \( h \to 0 \): \[ f'(x) = 4x^3 + 6x^2(0) + 4x(0)^2 + (0)^3 = 4x^3. \]
07
Domain of the Function and Its Derivative
The function \( f(x) = x^4 \) is a polynomial, and its domain is all real numbers, \( (-\infty, \infty) \). Similarly, its derivative \( f'(x) = 4x^3 \) is also a polynomial, so its domain is also all real numbers, \( (-\infty, \infty) \).
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.
Definition of Derivative
The derivative of a function at a specific point tells us how the function is changing at that point. It is essentially a slope of the tangent line to the function鈥檚 graph at a particular location. The formal definition of the derivative is expressed via a limit. For a function \( f(x) \), its derivative \( f'(x) \) is defined as:
\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\]This formula calculates the gradient by taking the limit of the average rate of change as the interval \( h \) approaches zero. This definition is central in calculus, offering a precise way to compute the instantaneous rate of change, which is crucial for analyzing curves.
\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\]This formula calculates the gradient by taking the limit of the average rate of change as the interval \( h \) approaches zero. This definition is central in calculus, offering a precise way to compute the instantaneous rate of change, which is crucial for analyzing curves.
Polynomial Functions
Polynomial functions are algebraic expressions that contain variables raised to whole number powers. A polynomial consists of terms in the form \( ax^n \), where \( a \) is a coefficient and \( n \) is a non-negative integer. The degree of a polynomial is determined by the highest power of the variable in the expression.
- The simplest polynomial is a constant, like \( f(x) = 5 \).
- A linear polynomial might look like \( f(x) = 3x + 2 \).
- A quadratic polynomial would be \( f(x) = x^2 - 4x + 4 \).
Binomial Theorem
The binomial theorem provides a way to expand expressions raised to a power, such as \((x + y)^n\). When using the binomial theorem, each term involves combinations of \( x \) and \( y \) raised to varying powers. The general formula is:
\[ (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\]In this formula:
\[ (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\]In this formula:
- \( \binom{n}{k} \) represents the binomial coefficient, which is calculated as \( \frac{n!}{k!(n-k)!} \).
- This coefficient determines how many ways we can choose \( k \) elements from \( n \) elements.
- Each term in the expansion derives from varying \( k \) from 0 to \( n \).
Domain of a Function
The domain of a function refers to all the possible input values (\( x \)-values) for which the function is defined. In simpler terms, it's all the numbers you can plug into the function without causing any mathematical mishaps, like division by zero.
- For most polynomial functions, including \( f(x) = x^4 \), the domain is all real numbers because you can insert any real number into the function and it will yield a valid output.
- This means the function is continuous across the entire real number line, \( (-\infty, \infty) \).
