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To find the derivative of \( f(x) = 5x^2 - 2x \) using first principles, we apply the definition of the derivative:

\[
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
\]

### Step 1: Find \( f(x+h) \)

We first need to compute \( f(x+h) \), where \( f(x) = 5x^2 - 2x \). So, substitute \( x+h \) into \( f(x) \):

\[
f(x+h) = 5(x+h)^2 - 2(x+h)
\]

Expand both terms:

\[
= 5(x^2 + 2xh + h^2) - 2(x + h)
= 5x^2 + 10xh + 5h^2 - 2x - 2h
\]

### Step 2: Compute \( f(x+h) - f(x) \)

Now, subtract \( f(x) = 5x^2 - 2x \) from \( f(x+h) \):

\[
f(x+h) - f(x) = (5x^2 + 10xh + 5h^2 - 2x - 2h) - (5x^2 - 2x)
\]

Simplifying:

\[
= 5x^2 + 10xh + 5h^2 - 2x - 2h - 5x^2 + 2x
\]
\[
= 10xh + 5h^2 - 2h
\]

### Step 3: Compute the derivative

Now, use the definition of the derivative:

\[
f'(x) = \lim_{h \to 0} \frac{10xh + 5h^2 - 2h}{h}
\]

Factor out \( h \) from the numerator:

\[
= \lim_{h \to 0} \frac{h(10x + 5h - 2)}{h}
\]

Cancel \( h \):

\[
= \lim_{h \to 0} (10x + 5h - 2)
\]

As \( h \to 0 \), the term \( 5h \) vanishes:

\[
f'(x) = 10x - 2
\]

Thus, the derivative of \( f(x) = 5x^2 - 2x \) is:

\[
f'(x) = 10x - 2
\]

### Do you want more details on any part of the process? Here are five related questions:

1. How is the first principles approach different from applying standard derivative rules?
2. Can the first principles method be applied to trigonometric functions? How?
3. What is the geometric interpretation of the derivative we found?
4. How would the process change if the function included higher-order terms like \( x^3 \)?
5. How can we verify the result using the power rule for derivatives?

**Tip:** The first principles method is foundational in calculus because it directly relies on the definition of the derivative, providing deeper insight into its meaning.

Find the derivative of f(x) = 5x^2 - 2x by first principles.

Learn how to find the derivative of the function f(x) = 5x^2 - 2x using the first principles approach. This problem explores key concepts in calculus, including the definition of derivatives. Suitable for students in Grades 11-12.

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