Find the derivative of y = x^2.

Step-by-step solution

1. 1

   Identify the form of the function. $y=x^2$ is a single power of $x$ with a constant exponent, which is exactly the case the power rule is built for. No product, quotient, or chain rule is needed.

2. 2

   State the power rule. For any constant exponent $n$,

   $\frac{d}{dx}x^n=n\,x^{\,n-1}$

   The exponent becomes the coefficient, and the new exponent is one lower.

3. 3

   Apply it with n = 2.

   $\frac{d}{dx}x^2=2\,x^{2-1}=2x^1=2x$

4. 4

   Confirm it from the limit definition. This is where the $2$ actually comes from:

   $\frac{(x+h)^2-x^2}{h}=\frac{x^2+2xh+h^2-x^2}{h}=\frac{2xh+h^2}{h}=2x+h$

   Letting $h\to0$ leaves $2x$ ✓. The $2xh$ cross-term of the binomial expansion is the entire source of the coefficient.

5. 5

   Interpret the result geometrically. The derivative $2x$ says the parabola's slope is $0$ at the vertex $x=0$, negative to the left, and positive to the right, growing linearly — which matches the shape of $y=x^2$.

6. 6

   Spot-check numerically. At $x=2.5$ the formula predicts a slope of $5$; the symmetric difference quotient $\frac{f(2.5001)-f(2.4999)}{0.0002}=5.0000$ ✓.