understanding the gradient of a curve through derivative

Question

Let's say I have a curve of the form $$y = x^2$$

therefore the gradient is $$\dfrac{dy}{dx} = 2x$$

My question is :

does this mean that the value of the gradient at $x = 3$ is $6$ or the change of $x$ by $3$ would give the value of the gradient as $6$?

Answer 1

The meaning here is that if you change $x$ by a tiny amount $h$ from the point $x=3$ to $x=3+h$, then the value of $y$ changes by $6h$. Go ahead and try it out for $h=0.1,0.01,0.001$. You will see why $h$ has to be small. The smaller $h$ is , the closer $\frac{\Delta y}{h}$ will be to the value 6.

At another point, say $x=4$, you find that changing $x$ from 4 to $4+h$ results in a change in $y$ of about $8h$. The scaling factor (6 or 8 etc) is what the derivative (gradient) gives you.

Answer 2

The definition of the derivative of a function is the rate of change of a function, say $f(x)$. It defines the gradient of the tangent at a specific point $x$. For a parabola, we visualise that the gradient is always changing, so the derivative of a parabola, say, $f(x)=x^2$ would be $f'(x)=2x$. The gradient of $f(x)$ at any point $x$ is $2x$.

So in summary, the derivative gives the gradient at point $x$.