Handle constants correctly

Stretching and lifting a curve

The faint curve is \(y = x^3\). The solid curve is \(y = cx^3 + k\), where the constants \(c\) and \(k\) are numbers that do not depend on \(x\). The solid curve is green where it rises and red where it falls. The orange tangent touches it at \(x = 1\), and the orange bar on the lower graph shows the slope of that tangent.

Slide \(c\). For \(c = 2\), every height doubles, and the slope at \(x = 1\) is 6. For \(c = -1\), the curve is reflected in the x-axis, and the slope is \(-3\). The slope of \(x^3\) at \(x = 1\) is 3, so the slope of the solid curve is \(c\) times 3.

Slide \(k\). The whole curve moves up or down, and the orange tangent moves with it without turning. Every point moves by the same amount, so the slope does not change, and the bar on the lower graph keeps its height.

Any curve

The faint curve is any curve \(f\), and the solid curve is \(cf + k\). The faint dashed line is the tangent to \(f\) at the same \(x\).

Multiplying by \(c\) multiplies every height by \(c\). A rise is a difference of two heights, so every rise, and therefore every slope, is multiplied by \(c\).

Adding \(k\) adds the same amount to every height, so \(k\) cancels in every rise, and the slope does not change.

At every \(x\), the slope of the solid curve is \(c\) times the slope of \(f\), for any \(k\).

Notation

1. A constant multiple stays in front, even when the constant is negative. The rise of \(cf\) is \(c\) times the rise of \(f\), so every difference quotient, and its limit, is multiplied by \(c\).

   \[\frac{d}{dx}\bigl[\,c\,f(x)\,\bigr] = \cy{c}\,f'(x)\]

2. A constant term has derivative 0, because its graph is a horizontal line, which has slope 0.

   \[\frac{d}{dx}\,k = 0\]

3. With the power rule, the constant \(c\) stays in front, and the exponent \(n\) multiplies it.

   \[\frac{d}{dx}\bigl[\,c\,x^n\,\bigr] = c\,n\,x^{n-1}\]

4. A number such as \(\pi\) or \(\sqrt{2}\) is a constant too, because it does not contain \(x\).

   \[\begin{aligned} \frac{d}{dx}\,\pi x^2 &= 2\pi x \\ \frac{d}{dx}\,\sqrt{2} &= 0 \end{aligned}\]

When it fails

The rule needs \(c\) to be a constant. If the factor contains \(x\), the factor changes as \(x\) changes. Let \(c = x\), so the curve is \(x \cdot x^3 = x^4\).

At \(x = 1\), copying the rule gives \(x \cdot 3x^2 = 3\), but the orange tangent has slope 4. The missing 1 comes from the change in the factor.

Example 1: a negative multiple

Find the derivative of \(y = -2x^4\).

1. The factor \(-2\) is a constant, so it stays in front. Differentiate \(x^4\) with the power rule. At \(x = 1\), the faint curve \(x^4\) has slope 4.

2. The exponent 4 comes down and multiplies \(-2\).

   \[y' = -2 \cdot 4x^3 = \cy{-8x^3}\]

3. The solid curve is stretched by 2 and reflected in the x-axis, so at \(x = 1\) its slope is \(-2 \cdot 4 = -8\). The curve is red there because it falls.

Example 2: a constant term

Find the derivative of \(y = 3x^2 + 5\). Then find the slope at \(x = 0\).

1. The curve is a sum of two terms, \(3x^2\) and 5. Differentiate each term.

2. A tempting wrong move: Keep the constant: the derivative of 5 is 5.

   \[y' = 6x + 5\]

   At \(x = 0\), this gives slope 5, the red line. But the lowest point of the curve is at \((0, 5)\), where the orange tangent is flat. The constant 5 only lifts the curve, so it cannot add slope.

3. The constant 5 has slope 0, because its graph is a horizontal line.

   \[y' = 6x + 0 = 6x\]

4. At \(x = 0\), the slope is \(6 \cdot 0 = 0\). This matches the flat tangent at the lowest point.

Practice

Find the derivative of \(y = \tfrac12 x^6 - 11\).

Slide the constant term up and down. The tangent does not turn, so the constant adds nothing to the slope.

Questions

Questions 1 and 2 use the graph of the solid curve \(y = cx^2 + k\) and the faint curve \(y = x^2\). Question 3 needs a calculation.

1. For \(y = 2x^2 - 3\), the faint curve has slope 2 at \(x = 1\). What is the slope of the solid curve at \(x = 1\)?

   • 4
   • 2
   • −1
   Answer

   4

2. The curve is now \(y = -2x^2 - 3\). What is its slope at \(x = 1\)?

   • −4
   • 4
   • −7
   Answer

   −4

3. Find the derivative of \(y = 6x^{1/2} + \pi\).

   • \(3x^{-1/2}\)
   • \(3x^{-1/2} + \pi\)
   • \(6x^{-1/2}\)
   • \(3x^{1/2}\)
   Answer

   \(3x^{-1/2}\)

Summary

You keep a constant multiple in front of the derivative, and you differentiate a constant term to 0.

This works because stretching a curve multiplies every slope by the same number, and lifting a curve does not change any slope.

You treat a number as a constant only if it does not contain \(x\).

This is one of four skills in topic 2.6 of Which & Why: AP Calculus.