Two Different Questions at the Same Input
For a function \(f\), the expression \(f(a)\) asks for the function’s output when its input is \(a\). The expression \(f'(a)\) asks a different question: what is the function’s instantaneous rate of change at that input?
Both expressions involve the same input \(a\), but they do not usually have the same value or meaning. The function value \(f(a)\) is the height of the graph at \(x=a\). When \(f\) is differentiable there, the derivative value \(f'(a)\) is the slope of the graph’s tangent line at that point.
What the Notation Means
The derivative of \(f\) at \(a\) is defined by the limit of the difference quotient, provided that limit exists:
This limit describes the instantaneous rate of change at \(a\). Equivalently, it is the limit of the slopes of secant lines as their second point approaches \((a,f(a))\). If the function is differentiable at \(a\), that limiting slope is the tangent slope there.
Writing \(f'(a)\) combines two operations. First, form the derivative of the function; then evaluate that derivative at the input \(a\). The same distinction appears in ordinary function notation: \(f(x)\) refers to the output at a general input \(x\), while \(f(a)\) refers to the output at one particular input.
The notation \(\frac{dy}{dx}\) identifies the derivative of \(y\) with respect to \(x\). The vertical evaluation mark, followed by \(x=a\), tells you where to evaluate it. Without that evaluation instruction, \(\frac{dy}{dx}\) can refer to the derivative at a general input; with the instruction, it refers to a particular derivative value.
Why the Two Derivative Notations Agree
Suppose \(y=f(x)\). The definition of the derivative at \(a\) uses the function’s values near \(a\) and the corresponding change in input. The notation \(f'(a)\) names the resulting derivative value directly. The Leibniz notation names the same rate as the derivative of \(y\) with respect to \(x\), then marks the input where that rate is evaluated.
Here \(y(x)=f(x)\), so the two difference quotients contain the same numerator and denominator. Their limits are therefore the same, when the limit exists. This is why the two notations can describe the same number without making \(f(a)\) and \(f'(a)\) the same thing.
Worked Examples
Worked Example: Read a Derivative from a Tangent Line
Suppose \(f(2)=5\), and the tangent line to the graph of \(y=f(x)\) at \(x=2\) passes through \((2,5)\) and \((4,-1)\). Find and interpret \(f(2)\), \(f'(2)\), and \(\left.\frac{dy}{dx}\right|_{x=2}\).
The function value is given directly:
The tangent slope is calculated from its two points. The change in vertical coordinate is \(-1-5=-6\), and the change in horizontal coordinate is \(4-2=2\). Therefore its slope is:
Because this line is tangent at \(x=2\), its slope is the derivative at that input. Thus \(f'(2)=-3\). Since \(y=f(x)\), the evaluated Leibniz notation has the same value:
Check the tangent-line information using point-slope form. A line with slope \(-3\) through \((2,5)\) is \(y-5=-3(x-2)\). At \(x=4\), this gives \(y-5=-6\), so \(y=-1\), as required. The point on the graph is \((2,5)\), while the tangent slope there is \(-3\): \(f(2)\) and \(f'(2)\) answer different questions.
Worked Example: Interpret Leibniz Notation at a Particular Input
Let \(y=f(x)\). Suppose the tangent line to the graph at \(x=-1\) is \(y=2x+1\). Determine \(f(-1)\), \(f'(-1)\), and \(\left.\frac{dy}{dx}\right|_{x=-1}\).
First, find the tangent point’s vertical coordinate by substituting \(x=-1\) into the tangent-line equation:
The tangent line touches the graph at that point, so the function value is \(f(-1)=-1\). The slope of the tangent line \(y=2x+1\) is its coefficient of \(x\), which is \(2\). Hence:
Because \(y=f(x)\), evaluating \(\frac{dy}{dx}\) at \(x=-1\) gives that same tangent slope:
Check that the tangent point lies on the given line: at \(x=-1\), the line gives \(y=-1\), matching \(f(-1)\). The function value is \(-1\); the derivative value is \(2\).
Worked Example: Distinguish Position from Instantaneous Velocity
Let \(s(t)\) represent an object’s position in feet after \(t\) seconds. At \(t=3\), suppose its position is \(11\) feet and the tangent line to the position graph is \(s=5t-4\). Interpret \(s(3)\) and \(\left.\frac{ds}{dt}\right|_{t=3}\).
The position at \(t=3\) is given as \(11\) feet. Check it against the tangent line’s point: substituting \(t=3\) gives:
Therefore \(s(3)=11\) feet. The slope of the tangent line \(s=5t-4\), with position on the vertical axis and time on the horizontal axis, is \(5\). Thus the instantaneous rate of change of position at \(t=3\) is:
This derivative is the instantaneous velocity at that time. The position is \(11\) feet; the rate is \(5\) feet per second. Check the tangent line’s slope by taking two points on it: at \(t=3\), \(s=11\); at \(t=4\), \(s=16\). Then \(\frac{16-11}{4-3}=5\), matching the derivative value.
A Reliable Reading Routine
When several derivative expressions appear together, do not treat them as interchangeable strings of symbols. Identify the object being evaluated and the input. The following routine keeps the meanings separate:
An expression such as \(f(a)\) asks for the output of \(f\) at input \(a\).
An expression such as \(f'(a)\) asks for the derivative value at \(a\), which is a rate and, geometrically, a tangent slope.
In \(\left.\frac{dy}{dx}\right|_{x=a}\), the condition \(x=a\) specifies where to evaluate the derivative.
When \(y=f(x)\), \(f'(a)\) and \(\left.\frac{dy}{dx}\right|_{x=a}\) name the same derivative value.
A Common Notation Mistake
A frequent error is to read \(f'(a)\) as though it meant “the value of \(f\) at \(a\),” and therefore set it equal to \(f(a)\). For example, in the first worked example, this mistaken reading would give:
That conclusion is wrong: the tangent slope is \(-3\), not \(5\). The mistake comes from overlooking the prime mark. The correct values are:
Another possible misreading is to treat the notation \(\left.\frac{dy}{dx}\right|_{x=a}\) as the function value at \(a\). The evaluation mark does not change the derivative into \(y\); it specifies the input at which the derivative is evaluated. When \(y=f(x)\), the result is \(f'(a)\), not \(f(a)\).
Check Your Understanding
For each question, keep the function value and derivative value distinct. The questions refer to a function \(f\) and, where stated, \(y=f(x)\).
- If \(f(4)=7\) and \(f'(4)=-2\), which number is the output at input \(4\), and which is the tangent slope there?
- When \(y=f(x)\), what does \(\left.\frac{dy}{dx}\right|_{x=3}\) represent?
- A tangent line at \(x=1\) has slope \(6\), and \(f(1)=-4\). State \(f'(1)\) and \(f(1)\), and explain their different meanings.
- An object’s position is \(s(2)=9\) feet, and its instantaneous velocity at \(t=2\) is \(3\) feet per second. Which notation represents the position, and which represents the velocity?
- True or false: \(f'(a)\) and \(f(a)\) must be equal because both expressions use the input \(a\). Explain briefly.
