LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
The derivative of f at x is the limit, as a step h shrinks to zero, of the change in f divided by h. That sentence is also a practical recipe. Keep h small but not zero, and the same ratio is a usable slope. For the cube, with h set to 0.0001, the ratio at 1 is 3.0003. The exact slope is 3. The lecture's point is that the leftover 0.0003 is often small enough, and that you never needed a differentiation rule to get it.
Why it matters
Rules such as the power rule, the product rule, and the chain rule are fast when they apply. They are a poor fit for a function you can only evaluate, which is the usual case in a simulation, a loss function, or a differential equation you intend to step forward in time. The definition asks only for two function values. That is why the lecture would rather see the limit used as a numerical tool than as an endless drill of formal proofs. Differentiability conditions are set aside. The cube is a polynomial, continuous and differentiable everywhere, so the assumption is safe for the example.
The idea in plain words
Apply the definition to g(x) = x squared. Form the rise over the run. Expand the shifted square, cancel the original square, and divide by the step. Send the step to zero and the slope is 2 times x. The power rule says the same thing. The algebra is there to show the two routes agree, not because you must repeat it for every new function. The numerical route stops before the limit. If the limit equals 2 times x, then for small h the ratio is about 2 times x. The lecture's money picture is the same idea: 2 dollars and 2.03 dollars are not equal, and for many decisions the 3 cents do not change the action. Error analysis can bound that gap. The lecture does not do the bound. It asks whether the direction of the slope is already clear. For h(x) = x cubed, refuse the power rule on purpose. Choose a step of 0.0001. Estimate the slope by cubing the shifted input, subtracting the cube of the current input, and dividing by the step. At 1, with a step of 0.0001, the lecture's calculator returns 3.0003. The power rule says the slope is 3 times x squared, hence 3 at x = 1. The two numbers differ in the fourth decimal. In a machine-learning step, what you often need is the direction to move.
How to use it
When you can write the derivative, use it and keep the finite-difference ratio as a check. When you cannot, pick a step, evaluate the function on either side or on one side as in the lecture's forward difference, and divide. Shrink the step if the answer is still moving. Do not shrink it to the point that subtraction of two nearly equal function values is swallowed by rounding. The lecture's 0.0001 is a demonstration at x = 1, not a universal step for every scale of x.
The same ratio is the opening move for numerical differential equations. A derivative is a slope. Replacing the slope by a difference quotient turns a differential equation into a stepping rule. Partial differential equations need the same replacement in more than one variable. The lecture stops at the single-variable definition and leaves those extensions for later. The cube example is enough to see that "derivative" and "small change over small step" are the same object at two levels of precision.
What usually goes wrong
Trusting a large step because the formula looks like the definition. The definition sends h to zero. A step of 0.1 on the cube at 1 gives (1.1 cubed minus 1) divided by 0.1, which is 3.31, not 3.0003. The approximation got worse because the step got lazy. The opposite error is demanding the exact 3 in a setting where 3.0003 would not change a decision. The lecture's question is which of those standards the job actually has.
A third error is applying the ratio at a kink or a jump and reading the output as a derivative. The lecture assumes the function is differentiable and says to set cusps aside. If the left-hand and right-hand ratios disagree, the limit is not there, and neither number is the slope.
Questions people ask
What exact inputs produced 3.0003?
The function x cubed, the point x = 1, and a forward step of 0.0001. The lecture evaluates 1.0001 cubed, subtracts 1, and divides by 0.0001.
Do I still need the power rule?
For hand calculation, yes. It gave 3 immediately. The numerical ratio is the fallback when the rule is unavailable or when you want a check that a coded derivative is not wildly off.
Is 3.0003 close enough?
The lecture says it depends on the job. For seeing which way a loss decreases, the extra 0.0003 did not flip the direction. A later use that needs several correct digits would take a smaller step or an exact derivative.