Introduction
A derivative tells us the slope of a tangent line at a point on a graph. Its sign and its magnitude convey different information: the sign distinguishes an upward-tilting tangent from a downward-tilting one, while the magnitude describes how steep the tangent is.
This distinction matters when comparing graphs. A derivative of \(-5\) is less than a derivative of \(2\), but its tangent is steeper in magnitude. To compare steepness, take the absolute value of each derivative; to describe the tangent’s direction, retain its sign.
Slope, Direction, and Steepness
For a line, slope is the change in vertical position divided by the change in horizontal position. A tangent line to the graph of a function has its own slope at the point of tangency. When \(f\) is differentiable at \(a\), the derivative \(f'(a)\) is that tangent slope.
The sign indicates the direction of the tangent as \(x\) increases. A positive slope tilts upward from left to right, and a negative slope tilts downward from left to right. The absolute value removes that directional information and leaves the slope’s size.
| Derivative value | Direction of tangent | Magnitude of slope |
|---|---|---|
| \(f'(a)=2\) | Upward from left to right | \(|f'(a)|=2\) |
| \(g'(b)=-2\) | Downward from left to right | \(|g'(b)|=2\) |
| \(h'(c)=-5\) | Downward from left to right | \(|h'(c)|=5\) |
The first two tangents have equal steepness by slope magnitude, even though they point in opposite directions. The third is steeper than either because its slope magnitude is larger. These comparisons concern tangent slopes, not whether one whole function is always steeper than another.
Worked Example: Comparing Tangents to Two Lines
Worked Example: Comparing Tangents to Two Lines
Compare the tangent slopes of \(f(x)=2x+1\) and \(g(x)=-3x+4\) at \(x=0\). Which tangent is steeper?
Use the derivative definition at \(a=0\):
For \(f(x)=2x+1\), \(f(h)=2h+1\) and \(f(0)=1\). Therefore, for \(h\ne 0\),
Thus \(f'(0)=\lim_{h\to0}2=2\). For \(g(x)=-3x+4\), \(g(h)=-3h+4\) and \(g(0)=4\). Hence
so \(g'(0)=\lim_{h\to0}(-3)=-3\). The magnitudes are \(|f'(0)|=|2|=2\) and \(|g'(0)|=|-3|=3\). The tangent to \(g\) is steeper, and it slopes downward from left to right; the tangent to \(f\) slopes upward.
Check: Both functions are lines, so each is its own tangent line. Their slopes are \(2\) and \(-3\), exactly the derivatives found from the limits.
Why the Absolute Value Measures Steepness
The derivative is defined using nearby secant slopes. At a differentiability point \(a\), the secant slope from \(x=a\) to \(x=a+h\) is \(\frac{f(a+h)-f(a)}{h}\), for nonzero \(h\). As \(h\) approaches zero, these slopes approach \(f'(a)\). Taking absolute values turns each signed secant slope into its magnitude.
Justification. Let \(s(h)=\frac{f(a+h)-f(a)}{h}\) for \(h\ne0\). Differentiability at \(a\) means \(\lim_{h\to0}s(h)=f'(a)\). For any real numbers \(u\) and \(v\), the reverse triangle inequality gives \(\big||u|-|v|\big|\le |u-v|\). Apply it with \(u=s(h)\) and \(v=f'(a)\):
The right-hand side approaches zero because \(s(h)\) approaches \(f'(a)\). The left-hand side must therefore approach zero as well, which proves that \(|s(h)|\) approaches \(|f'(a)|\). Thus the absolute derivative records the limiting size of nearby slopes, without their direction.
Worked Example: Comparing Slopes on One Curve
Worked Example: Comparing Slopes on One Curve
For \(p(x)=x^2\), compare the tangent steepness at \(x=-1\) and \(x=2\), using first principles.
At an arbitrary input \(a\), \(p(a+h)=(a+h)^2=a^2+2ah+h^2\) and \(p(a)=a^2\). The difference quotient is
Taking \(h\to0\) gives \(p'(a)=2a\). At \(a=-1\), \(p'(-1)=2(-1)=-2\), so the slope magnitude is \(2\). At \(a=2\), \(p'(2)=2(2)=4\), so the slope magnitude is \(4\). Therefore the tangent at \(x=2\) is steeper, and the two tangents point in opposite directions.
We can also write the tangent lines. At \((-1,1)\), the slope is \(-2\), so \(y-1=-2(x+1)\), or \(y=-2x-1\). At \((2,4)\), the slope is \(4\), so \(y-4=4(x-2)\), or \(y=4x-4\).
Check: Substituting \(x=-1\) into \(y=-2x-1\) gives \(y=2-1=1=p(-1)\), and its slope is \(-2\). Substituting \(x=2\) into \(y=4x-4\) gives \(y=8-4=4=p(2)\), and its slope is \(4\). The tangent equations agree with the contact points and derivative values.
Compare Magnitudes, Not Signed Values
Signed numerical order and steepness are different comparisons. The statement \(-5<2\) compares the derivative values as real numbers. It does not say that slope \(-5\) is less steep than slope \(2\). Their magnitudes are \(5\) and \(2\), so slope \(-5\) is steeper.
Worked Example: Avoiding a Signed-Value Comparison
Suppose two differentiable functions have derivative values \(q'(a)=-4\) and \(r'(b)=3\). Which tangent is steeper, and what are their directions?
First compare magnitudes:
The tangent to \(q\) is steeper. Since \(q'(a)=-4<0\), it slopes downward as the input increases. Since \(r'(b)=3>0\), the tangent to \(r\) slopes upward.
Check: The signed values retain the opposite directions, while their absolute values compare steepness. The conclusion follows from \(4>3\), not from comparing \(-4\) and \(3\) in signed order.
A Common Pitfall: Treating a Negative Slope as a Small Steepness
A common error is to reason that a negative derivative must represent a less steep tangent than any positive derivative. For example, someone might write
The first inequality is true, but the conclusion does not follow: it compares signed values rather than steepness. The correction is to compare absolute values:
Therefore slope \(-5\) is steeper than slope \(2\), but points in the opposite direction. Use the derivative’s sign for direction and its magnitude for steepness.
Interpreting a Graph Carefully
A graph gives a visual impression of tangent steepness, but that impression is meaningful for comparisons only when the horizontal and vertical axes use consistent scales and units. Stretching one axis relative to the other changes how steep a line appears on the page. The derivative remains the slope measured in the graph’s coordinate units; the drawing alone can mislead if the scales are not comparable.
Also, a derivative describes a tangent at a particular input. It does not, by itself, describe the steepness at every point or the overall shape of an entire curve. To make a local comparison, identify the input, compare the derivative magnitudes there, and use the signs separately to describe direction.
Check Your Understanding
Use the sign and magnitude of a derivative as separate pieces of information.
- Two tangent slopes are \(6\) and \(-6\). How do their directions and steepness compare?
- Which tangent is steeper, one with slope \(-7\) or one with slope \(4\)? State the direction of each.
- If a tangent slope has magnitude \(1/3\), what are the possible signed slope values?
- For \(p(x)=x^2\), use the first-principles difference quotient to find the tangent slope at \(x=1\), then state its magnitude.
- Why can different horizontal and vertical graph scales affect the visual impression of steepness?
