Issues with derivative

Question

I'm a beginner at Calculus. So, don't laugh at my question. We know that $$\dfrac{d}{dx} (x^2) = 2x.$$ If $x=4$, then the value of the function $y=x^2$ becomes $16$. So shouldn't the slope be $$\dfrac{(16-0)}{(4-0)}=4\ ?$$ But $2x=8$ [assuming $x=4$].

Answer 1

You're using $\frac{16-0}{4-0}$ to calculate the gradient of the line, however this is only valid if it were a straight line from $x=0$ to $x=4$. As the line $y=x^2$ is curved, the $\Delta y$ and $\Delta x$ needs to be infinitesimally small, and hence we use $\frac{dy}{dx}$.

Answer 2

The problem is that you’ve used two points the calculate slope—one of which is the origin. This leads to an inaccurate conclusion about the derivative.

You can think of this as anchoring the tangent line to two points. For the best tangent line, you want it anchored to only one point via the derivative.