https://r2.erweima.ai/mathbot/image/172534386245901lfe0vu.jpg

The function given in the image is:

\[
G(x) = \frac{2x}{x^2 - x}
\]

To find the derivative \( G'(x) \) using the definition of the derivative, we use the formula:

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

Let's compute \( G'(x) \) step-by-step.

### Step 1: Compute \( G(x+h) \)

First, find the expression for \( G(x+h) \):

\[
G(x+h) = \frac{2(x+h)}{(x+h)^2 - (x+h)}
\]

Expand \( (x+h)^2 \) and simplify:

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

So,

\[
G(x+h) = \frac{2(x+h)}{x^2 + 2xh + h^2 - x - h}
\]

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

Now, compute \( G(x+h) - G(x) \):

\[
G(x+h) - G(x) = \frac{2(x+h)}{x^2 + 2xh + h^2 - x - h} - \frac{2x}{x^2 - x}
\]

To combine these into a single fraction, find the common denominator:

\[
\text{Common denominator} = (x^2 + 2xh + h^2 - x - h)(x^2 - x)
\]

Now, express the difference:

\[
G(x+h) - G(x) = \frac{2(x+h)(x^2 - x) - 2x(x^2 + 2xh + h^2 - x - h)}{(x^2 + 2xh + h^2 - x - h)(x^2 - x)}
\]

### Step 3: Simplify and find the limit

Expand and simplify the numerator, then divide by \( h \) and take the limit as \( h \to 0 \):

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

Factor out terms, simplify, and take the limit to find \( G'(x) \). This process involves algebraic manipulation to cancel out \( h \) in the numerator.

Let's proceed with simplifying the above expression. This requires some careful algebraic steps, which I'll do now.

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

Simplify and factor out \( h \) from the numerator and denominator:

\[
G'(x) = \lim_{h \to 0} \frac{2(-h^2)}{h(x^2 - x)(x^2 - x + h)}
\]

As \( h \to 0 \):

\[
G'(x) = \frac{2x(x - 1) - 2x(x - 1)}{(x^2 - x)(x^2 - x)}
\]

Thus, the derivative \( G'(x) \) simplifies to:

\[
G'(x) = \frac{2(x^2 - x) - 2x^2 + x}{(x^2 - x)^2}
\]

After cancellation and further simplification, you will find:

\[
G'(x) = \frac{2(x^2 - x)}{(x^2 - x)^2} = \frac{2}{x^2 - x}
\]

This final result shows the derivative of the function \( G(x) \) using the definition of the derivative. 

Would you like more details on any step? 

Here are some related questions to explore further:

1. How do you compute derivatives using the quotient rule instead of the definition?
2. How do limits work in calculus, especially when finding derivatives?
3. Can the same derivative be found using other methods, such as implicit differentiation?
4. How does one simplify complex rational expressions effectively?
5. What are the conditions for a function to be differentiable at a point?

**Tip:** Always check if you can simplify an expression before taking limits; this often makes calculations easier.

Aprende cómo calcular la derivada de una función utilizando la definición de derivada. Este ejercicio resuelto abarca los conceptos clave de derivadas y teoremas límite. Adecuado para estudiantes de Grados 11-12.

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