📚 Topic Summary
The Power Rule and Constant Multiple Rule are fundamental tools in differential calculus. The Power Rule helps us find the derivative of terms in the form $x^n$, where $n$ is any real number. It states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$. The Constant Multiple Rule simplifies finding the derivative of a constant multiplied by a function. It says that if $f(x) = c \cdot g(x)$, where $c$ is a constant, then $f'(x) = c \cdot g'(x)$. Together, these rules allow us to differentiate a wide range of functions efficiently.
🧠 Part A: Vocabulary
Match the term with its definition:
| Term |
Definition |
| 1. Power Rule |
a. A number multiplied by a variable. |
| 2. Constant Multiple Rule |
b. The process of finding the derivative of a function. |
| 3. Derivative |
c. Used to find the derivative of $x^n$. |
| 4. Constant |
d. A fixed number. |
| 5. Coefficient |
e. Used to find the derivative of $c \cdot f(x)$. |
✏️ Part B: Fill in the Blanks
Complete the following paragraph with the correct words:
The _________ Rule states that the derivative of $x^n$ is $n$ times $x$ to the power of _________. The _________ Multiple Rule tells us that the derivative of a constant times a function is the _________ times the derivative of the function.
🤔 Part C: Critical Thinking
Explain in your own words how the Power Rule and Constant Multiple Rule can be used together to differentiate a polynomial function.