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The question asks to choose the correct form of the derivative of the function \( f(x) = 5x^2 + 3x - 2 \) from the definition of the derivative. The definition of the derivative is given by:

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

For the function \( f(x) = 5x^2 + 3x - 2 \), let's first calculate \( f(x+h) \):

\[
f(x+h) = 5(x+h)^2 + 3(x+h) - 2
\]
Expanding the terms:

\[
f(x+h) = 5(x^2 + 2xh + h^2) + 3x + 3h - 2
\]
\[
f(x+h) = 5x^2 + 10xh + 5h^2 + 3x + 3h - 2
\]

Now, subtract \( f(x) = 5x^2 + 3x - 2 \):

\[
f(x+h) - f(x) = (5x^2 + 10xh + 5h^2 + 3x + 3h - 2) - (5x^2 + 3x - 2)
\]
Simplifying the expression:

\[
f(x+h) - f(x) = 10xh + 5h^2 + 3h
\]

Now, substitute this result into the definition of the derivative:

\[
f'(x) = \lim_{h \to 0} \frac{10xh + 5h^2 + 3h}{h}
\]

Simplify the fraction:

\[
f'(x) = \lim_{h \to 0} \frac{h(10x + 5h + 3)}{h}
\]

Cancel out the \( h \):

\[
f'(x) = \lim_{h \to 0} (10x + 5h + 3)
\]

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

\[
f'(x) = 10x + 3
\]

Now, let's look at the options provided. The correct answer should match the expression for \( \lim_{h \to 0} \frac{10xh + 5h^2 + 3h}{h} \). Option **G** matches this result.

Do you want further details or have any questions?

### 5 Related Questions:
1. What is the purpose of the definition of a derivative in calculus?
2. How do you expand \( (x+h)^2 \) when applying the definition of the derivative?
3. What happens to higher-order \( h \)-terms as \( h \to 0 \)?
4. Why do we cancel the \( h \) in the limit process?
5. How would this process change for a cubic function?

### Tip:
Always carefully expand and simplify when applying the definition of a derivative to ensure all terms are correctly accounted for.

Choose the correct form of the derivative of 5x^2 + 3x - 2 from the definition of derivative.

Learn how to find the derivative of the function 5x^2 + 3x - 2 using the definition of the derivative. This solution walks through key calculus concepts such as limits and the definition of a derivative, and provides step-by-step calculations suitable for high school calculus students.

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