what is the derivative of  h(x) = x²+ 4x – 5 dan g(x) = 4x - 4

To find the derivatives of the functions \( h(x) = x^2 + 4x - 5 \) and \( g(x) = 4x - 4 \), we'll use basic differentiation rules.

1. **Derivative of \( h(x) = x^2 + 4x - 5 \):**

   The derivative of \( h(x) \) with respect to \( x \) is found by differentiating each term:

   \[
   h'(x) = \frac{d}{dx}(x^2) + \frac{d}{dx}(4x) - \frac{d}{dx}(5)
   \]

   \[
   h'(x) = 2x + 4
   \]

   So, the derivative of \( h(x) \) is \( h'(x) = 2x + 4 \).

2. **Derivative of \( g(x) = 4x - 4 \):**

   The derivative of \( g(x) \) with respect to \( x \) is:

   \[
   g'(x) = \frac{d}{dx}(4x) - \frac{d}{dx}(-4)
   \]

   \[
   g'(x) = 4
   \]

   So, the derivative of \( g(x) \) is \( g'(x) = 4 \).

**Summary:**
- \( h'(x) = 2x + 4 \)
- \( g'(x) = 4 \)

Would you like any further explanation, or is there another aspect of these functions you're curious about?

Here are some related questions you might consider:

1. What are the critical points of \( h(x) \) based on its derivative?
2. How do the derivatives \( h'(x) \) and \( g'(x) \) affect the graph of each function?
3. What is the second derivative of \( h(x) \)?
4. Where does \( h'(x) = g'(x) \)?
5. What is the slope of the tangent line to \( h(x) \) at \( x = 2 \)?
6. How does the derivative \( h'(x) \) compare to the slope of \( h(x) \) at various points?
7. How do the derivatives help in determining the increasing or decreasing nature of the functions?
8. How would you use these derivatives to solve optimization problems?

**Tip:** The derivative of a function gives the slope of the tangent line at any point on the curve, which can help in understanding the behavior of the function (increasing or decreasing) at that point.

Learn how to find derivatives of functions h(x) = x^2 + 4x - 5 and g(x) = 4x - 4 with step-by-step solutions. This problem covers differentiation using the power rule and its application to polynomial and linear functions. Suitable for Grades 11-12.

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