Question

By means of the general theorem (8) write down the derivatives of the following functions: $$ \begin{array}{lllll} x^{4} ; & x^{5} ; & x^{10} ; & x ; & x^{\infty 9} \end{array} $$

Short answer

Question: Use the Power Rule to find the derivatives of the following functions: 1. \(f(x) = x^4\) 2. \(f(x) = x^5\) 3. \(f(x) = x^{10}\) 4. \(f(x) = x\) Answer: 1. \(f'(x) = 4x^3\) 2. \(f'(x) = 5x^4\) 3. \(f'(x) = 10x^9\) 4. \(f'(x) = 1\)

Step-by-step solution

Function 1: \(x^4\)

To find the derivative of the function \(f(x) = x^4\), we apply the Power Rule: $$f'(x) = nx^{n-1}$$ Here, \(n = 4\), so: $$f'(x) = 4x^{4-1} = 4x^3$$

Function 2: \(x^5\)

To find the derivative of the function \(f(x) = x^5\), we apply the Power Rule: $$f'(x) = nx^{n-1}$$ Here, \(n = 5\), so: $$f'(x) = 5x^{5-1} = 5x^4$$

Function 3: \(x^{10}\)

To find the derivative of the function \(f(x) = x^{10}\), we apply the Power Rule: $$f'(x) = nx^{n-1}$$ Here, \(n = 10\), so: $$f'(x) = 10x^{10-1} = 10x^9$$

Function 4: \(x\)

To find the derivative of the function \(f(x) = x\), we apply the Power Rule: $$f'(x) = nx^{n-1}$$ Here, \(n = 1\), so: $$f'(x) = 1x^{1-1} = 1x^0 = 1$$

Function 5: \(x^{9\infty}\)

Since this function is not defined properly, it does not have a meaningful derivative. Please check the function to make sure it is correctly written.

Key concepts

Power Rule Differentiation

Power Rule differentiation is a fundamental technique in calculus for finding the derivative of power functions, where a variable is raised to a constant power. The Power Rule states that if you have a function of the form
\( f(x) = x^n \)
then its derivative
\( f'(x) \)
is given by
\( f'(x) = nx^{n-1} \).
To apply this rule, you multiply the original power by the coefficient and subtract one from the power. For example, for the function \( f(x) = x^4 \), we use the Power Rule to find the derivative
\( f'(x) = 4x^{4-1} = 4x^3 \).
The beauty of the Power Rule lies in its simplicity and its application to any real-numbered power, making it a powerful tool in the arsenal of calculus techniques.

Calculus Education

Calculus education aims to equip students with a strong understanding of the concepts and techniques of differentiation and integration. Essential to this education is the practice of applying differentiation rules, like the Power Rule, and understanding when and how to use them. Educators emphasize the importance of recognizing different types of functions and selecting the appropriate rule to differentiate them.

Interactive Learning

Through interactive problems, students can see these rules in action, which greatly enhances their learning experience. A solid calculus education prepares students not only for academic success but also for real-world problems where mathematical modeling and solutions are required.

Basic Derivative Rules

When beginning to learn calculus, students are introduced to basic derivative rules which serve as the building blocks for more complex calculations. These include the Power Rule, the Constant Rule (the derivative of any constant is zero), and the Sum Rule (the derivative of a sum is the sum of the derivatives).

Applying Multiple Rules

For more complex functions, it may be necessary to combine these rules to find the derivative. For instance, for a polynomial like \( f(x) = 3x^5 + 2x - 7 \), we would apply the Power Rule to each term involving a power of \( x \) and the Constant Rule to differentiate the constant term.

Higher Order Derivatives

Higher Order Derivatives refer to derivatives taken multiple times. The second derivative, for example, is the derivative of the first derivative, and provides information about a function's curvature or concavity. This concept is crucial for understanding motion in physics, where the second derivative is related to acceleration.

General Method

The same rules used for first derivatives also apply when taking higher order derivatives. If we revisit our earlier example \( f(x) = x^4 \) with a first derivative \( f'(x) = 4x^3 \), the second derivative would be \( f''(x) = 12x^2 \), again using the Power Rule. Higher order derivatives often arise in Taylor series expansions and differential equations, making it another significant concept in calculus.