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+---
+title: Calculating a Derivative by Definition
+description: Suppose \$ f(x) = ax^2 + bx + c \$ (you can set \$ a=1, b=0, c=0 \$ for \$ f(x) = x^2 \$, but we’ll keep it general). $$ f(x + h) = a(x + h)^2 + b(x + h) + c = a(x^2 + 2xh + h^2) + b(x + h) + c $$ $$ = ax^2 + 2axh + ah^2 + bx + bh + c $$ $$ f(x) = ax^2 + bx + c $$
+tags: 
+- mathematics
+categories:
+- graded assignment
+image: /images/tree.jpg
+excludeSearch: false
+width: wide
+---
+
+## **Step-by-Step: Calculating a Derivative by Definition**
+
+Let’s use the **four-step process** for finding the derivative of a simple quadratic function, as this approach is both fundamental and widely used in calculus[^3][^4]. For this example, let’s use a general quadratic function, but if you want a specific example from the PDF, we’ll use the quadratic function \$ f(x) = x^2 \$ (which is used in the context of quadratic functions[^1]).
+
+However, the PDF also discusses the slope of a quadratic function \$ f(x) = ax^2 + bx + c \$, mentioning that the slope at any point is \$ 2ax + b \$. Let’s see how this is derived using the definition.
+
+---
+
+### **Step 1: Write \$ f(x + h) \$ and \$ f(x) \$**
+
+Suppose \$ f(x) = ax^2 + bx + c \$ (you can set \$ a=1, b=0, c=0 \$ for \$ f(x) = x^2 \$, but we’ll keep it general).
+
+$$
+f(x + h) = a(x + h)^2 + b(x + h) + c = a(x^2 + 2xh + h^2) + b(x + h) + c
+$$
+
+$$
+= ax^2 + 2axh + ah^2 + bx + bh + c
+$$
+
+$$
+f(x) = ax^2 + bx + c
+$$
+
+---
+
+### **Step 2: Compute \$ f(x + h) - f(x) \$**
+
+Subtract \$ f(x) \$ from \$ f(x + h) \$:
+
+$$
+f(x + h) - f(x) = (ax^2 + 2axh + ah^2 + bx + bh + c) - (ax^2 + bx + c)
+$$
+
+$$
+= 2axh + ah^2 + bh
+$$
+
+$$
+= h(2ax + ah + b)
+$$
+
+---
+
+### **Step 3: Divide by \$ h \$**
+
+$$
+\frac{f(x + h) - f(x)}{h} = \frac{h(2ax + ah + b)}{h} = 2ax + ah + b
+$$
+
+---
+
+### **Step 4: Take the limit as \$ h \to 0 \$**
+
+$$
+f'(x) = \lim_{h \to 0} (2ax + ah + b) = 2ax + b
+$$
+
+---
+
+## **Summary Table**
+
+| Step | Action | Expression/Result |
+| :-- | :-- | :-- |
+| 1 | Write \$ f(x + h) \$, \$ f(x) \$ | \$ f(x+h) = a(x+h)^2 + ··· \$ |
+| 2 | Subtract \$ f(x) \$ from \$ f(x+h) \$ | \$ h(2ax + ah + b) \$ |
+| 3 | Divide by \$ h \$ | \$ 2ax + ah + b \$ |
+| 4 | Take the limit \$ h \to 0 \$ | \$ 2ax + b \$ |
+
+
+---
+
+## **Example from the PDF**
+
+The PDF states (in Section 3.2):
+> “For the quadratic function described as \$ f(x) = ax^2 + bx + c \$ where \$ a \neq 0 \$, the slope of \$ f \$ at any given point \$ (x, f(x)) \$ is \$ (2ax + b) \$.”
+
+This matches exactly the result we derived above using the definition of the derivative[^1].
+
+---
+
+## **Quick Example: \$ f(x) = x^2 \$**
+
+Let’s do the calculation for \$ f(x) = x^2 \$:
+
+1. **Step 1:**
+\$ f(x + h) = (x + h)^2 = x^2 + 2xh + h^2 \$
+\$ f(x) = x^2 \$
+2. **Step 2:**
+\$ f(x + h) - f(x) = 2xh + h^2 \$
+3. **Step 3:**
+\$ \frac{f(x + h) - f(x)}{h} = \frac{2xh + h^2}{h} = 2x + h \$
+4. **Step 4:**
+\$ \lim_{h \to 0} (2x + h) = 2x \$
+
+So, the derivative of \$ f(x) = x^2 \$ is \$ f'(x) = 2x \$, which matches the general result above (with \$ a=1, b=0 \$)[^1][^3].
+
+---
+
+## **Related Question**
+
+**Q:** What is the slope of the quadratic function \$ f(x) = 3x^2 + 4x + 1 \$ at \$ x = 2 \$?
+
+**A:**
+Using the general result \$ f'(x) = 2ax + b \$,
+\$ a = 3, b = 4 \$, so
+\$ f'(x) = 6x + 4 \$.
+At \$ x = 2 \$:
+\$ f'(2) = 6 \times 2 + 4 = 16 \$.
+
+---
+
+## **Summary**
+
+- **The derivative is the slope of the tangent to the curve at a point.**
+- **Use the four-step process:**
+
+1. Write \$ f(x + h) \$ and \$ f(x) \$.
+2. Compute \$ f(x + h) - f(x) \$.
+3. Divide by \$ h \$.
+4. Take the limit as \$ h \to 0 \$.
+- **For quadratic functions, the derivative is \$ 2ax + b \$.**
+
+This method works for any differentiable function, and the PDF confirms this for quadratic functions[^1][^3].
+
+<div style="text-align: center">⁂</div>
+
+[^1]: M1_VOL1_SETS-FUNCTIONS.pdf
+
+[^2]: https://www.whitman.edu/mathematics/calculus/calculus_03_Rules_for_Finding_Derivatives.pdf
+
+[^3]: https://math.wvu.edu/~hlai2/Teaching/Tip-Pdf/Tip1-5.pdf
+
+[^4]: https://faculty.ung.edu/jallagan/Courses materials/Math 1450 Calculus 1/Syllabus and ebook/problems and solutions for calculus 1.pdf
+
+[^5]: https://www2.math.binghamton.edu/lib/exe/fetch.php/people/mckenzie/derivative_sections_in_bittinger.pdf
+
+[^6]: https://tutorial.math.lamar.edu/pdf/calculus_cheat_sheet_derivatives.pdf
+
+[^7]: https://ocw.mit.edu/ans7870/textbooks/Strang/Edited/Calculus/2.1-2.4.pdf
+
+[^8]: https://mathguy.us/Handbooks/CalculusHandbook.pdf
+
+[^9]: https://people.math.wisc.edu/~angenent/Free-Lecture-Notes/free221.pdf
+
+[^10]: https://www.scribd.com/document/598700870/Calculus-Book
+