The variable changes exactly as fast as itself.
If \(f(x) = x\), then increasing \(x\) by \(h\) increases the value by exactly \(h\). The difference quotient is \(\frac{(x + h) - x}{h} = \frac{h}{h} = 1\) for every \(h \neq 0\), so the derivative is \(1\).
This is the base case of the whole process. Every other rule reduces a complicated expression to the derivatives of simpler pieces; the recursion has to stop somewhere, and it stops here and at the constant rule.
It is also the \(n = 1\) case of the power rule: \(\frac{d}{dx}x^{1} = 1 \cdot x^{0} = 1\).
Every line is the step the calculator would show, in the order it applies them. Each graph is live: hover it to read both curves and see the tangent whose slope is the derivative, drag to pan, scroll to zoom.
The expression is the variable itself.
f is the 45° line and f′ is the constant 1. Read the pair vertically — the height of f′ at any point is the steepness of f directly above it, and that steepness never changes. Hover anywhere and f′ reads exactly 1.
The line \(y = x\) rises one unit for every unit across — slope \(1\).
Open x in the calculatorHere the differentiation variable is \(t\), and the expression is \(t\).
The identical picture with t on the axis instead of x. The letter is irrelevant; the expression matches the variable being differentiated.
The letter does not matter, only whether it matches the variable you are differentiating with respect to.
Open t in the calculatorSplit the sum.
The variable rule gives \(1\); the constant rule gives \(0\).
The 45° line lifted so that it crosses the axis at \(x = -5\), with f′ still pinned at 1 — the two lines are parallel everywhere.
Adding a constant shifts the graph up or down without tilting it, so the slope is unchanged.
Open x + 5 in the calculatorPull the constant factor out.
Then the variable rule.
Five times the steepness, and f′ answers with the constant 5. Its height has changed while its shape has not.
The \(1\) is where the answer comes from, even though it is invisible in the final \(5\).
Open 5x in the calculatorProduct rule, with \(f = x\) and \(g = \sin x\).
The variable rule supplies the first derivative, the trigonometric rule the second.
Here the variable rule is only one factor inside a product: f oscillates with a growing swing, and f′ crosses zero at each crest and trough.
In real problems the variable rule almost always appears as a factor of \(1\) inside a bigger rule.
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