why didn't use the power rule for -3x and how is derivative of 1 is 0

Great questions! Let's address each one:

### 1. Why didn't we use the power rule for \(-3x\)?

The power rule states that the derivative of \( x^n \) is \( nx^{n-1} \). While we can technically apply the power rule to \( -3x \), it's just a special case of the power rule where \( n = 1 \).

Given \( -3x \):

\[
-3x = -3x^1
\]

Applying the power rule:

\[
\frac{d}{dx}(-3x^1) = -3 \cdot 1 \cdot x^{1-1} = -3 \cdot x^0 = -3 \cdot 1 = -3
\]

So, we did use the power rule; it's just simpler in this case, which is why it might not seem like we did.

### 2. How is the derivative of 1 equal to 0?

The derivative of a constant is always zero. This is because the derivative measures the rate of change of a function. A constant function, such as \( f(x) = 1 \), does not change no matter what \( x \) is.

Mathematically:

If \( c \) is a constant, then:

\[
\frac{d}{dx}(c) = 0
\]

For \( f(x) = 1 \):

\[
\frac{d}{dx}(1) = 0
\]

This is because a constant value has no variation with respect to \( x \); it's always the same, so its rate of change (derivative) is zero.

Would you like more details or have any questions?

Here are some related questions to expand your understanding:

1. How do we apply the power rule to find the derivative of \( x^3 \)?
2. What is the derivative of a sum of functions, like \( x^2 + 3x - 5 \)?
3. Can we use the power rule to find the derivative of a negative exponent, like \( x^{-2} \)?
4. What is the general rule for the derivative of a constant multiplied by a function, like \( 4x^3 \)?
5. How do we find the derivative of a polynomial function?
6. What is the chain rule in differentiation, and when is it used?
7. How do we differentiate a function that is a product of two functions, like \( x^2 \cdot e^x \)?
8. How does the concept of the derivative relate to the slope of the tangent line at a point on a graph?

**Tip:** Understanding the rules of differentiation, like the power rule, product rule, and chain rule, is essential for solving more complex calculus problems. Make sure to practice applying these rules in various contexts to strengthen your skills.

Explore why the power rule applies to -3x and how the derivative of 1 is 0. This page provides detailed explanations and examples of differentiation rules, suitable for students in Grades 11-12.

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