Differentiate y = x squared.

Step-by-step solution

1. 1

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

   $\frac{d}{dx}\left(x^n\right)=nx^{n-1}$

   Two things happen at once: the old exponent comes down in front as a multiplier, and the exponent itself drops by one.

2. 2

   Identify the exponent. In $y=x^2$ we have $n=2$. There is no coefficient to carry and no chain rule needed, since the base is simply $x$.

3. 3

   Apply the rule.

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

4. 4

   Confirm from the definition, so the rule is not just a recipe. The difference quotient is

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

   and letting $h\to 0$ gives $2x$ — the same answer, obtained without assuming the power rule.

5. 5

   Read the meaning off the graph. The derivative $2x$ is the slope of the parabola: at $x=3$ the tangent has slope $6$, at $x=0$ it is flat (the vertex), and for $x<0$ the slope is negative. So the steepness grows linearly as you move away from the origin.