The Derivative and Its Definition: math lesson, College

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Math lessons College : The Derivative and Its Definition — Zyro the alien explorer of Planète Maths

A speedometer does not tell you your average speed over a trip. It tells you how fast you are going right now. Calculus makes that idea precise: the derivative measures the instantaneous rate of change of a quantity, and it also gives the slope of a curve at a single point. In this chapter you will build the derivative from a limit, use it to write tangent lines, and learn the first rules that make computing derivatives fast.

Sommaire

1. Average rate of change and the difference quotient

Suppose a function \(f\) describes some quantity, such as the height of a drone or the cost of producing items. Between \(x=a\) and \(x=a+h\), the quantity changes from \(f(a)\) to \(f(a+h)\).

Definition: difference quotient

For \(h\neq 0\), the difference quotient of \(f\) at \(a\) is
\[\dfrac{f(a+h)-f(a)}{h}.\]
It is the average rate of change of \(f\) on the interval between \(a\) and \(a+h\). Geometrically, it is the slope of the secant line through the points \((a,f(a))\) and \((a+h,f(a+h))\).

Example 1: a secant slope

Let \(f(x)=x^2\) and \(a=1\). Then
\[\dfrac{f(1+h)-f(1)}{h}=\dfrac{(1+h)^2-1}{h}=\dfrac{2h+h^2}{h}=2+h.\]
For \(h=2\), point \(B\) is \((3,9)\) and the secant slope is \(4\), as in the figure. Watch what happens when \(h\) shrinks:

h	Point B	Slope of secant AB = 2 + h
1	(2, 4)	3
0.1	(1.1, 1.21)	2.1
0.01	(1.01, 1.0201)	2.01
0.001	(1.001, 1.002001)	2.001

The slopes get closer and closer to \(2\).

2. Instantaneous rate of change: letting h shrink

The table suggests a number the slopes are heading toward. To capture an instantaneous rate we let \(h\) approach \(0\) without ever setting \(h=0\) (that would give \(0/0\)). The limit of the average rates is the instantaneous rate of change.

Common mistake

You cannot plug \(h=0\) into the difference quotient directly. First simplify algebraically (expand, factor, cancel \(h\)), and only then take the limit.

3. The derivative as a limit

Definition: derivative

The derivative of \(f\) at \(a\) is
\[f'(a)=\lim_{h\to 0}\dfrac{f(a+h)-f(a)}{h},\]
when this limit exists. If it exists, \(f\) is differentiable at \(a\). The derivative function is \(f'(x)=\lim_{h\to 0}\dfrac{f(x+h)-f(x)}{h}\). Other notations: \(\dfrac{dy}{dx}\) and \(\dfrac{d}{dx}f(x)\).

The number \(f'(a)\) has two meanings: the slope of the tangent line at \((a,f(a))\), and the instantaneous rate of change of \(f\) at \(a\), measured in units of \(f\) per unit of \(x\) (for example feet per second).

Example 2: derivative from the definition

Let \(f(x)=3x^2-5x\). Then
\[\dfrac{f(x+h)-f(x)}{h}=\dfrac{3(x^2+2xh+h^2)-5x-5h-3x^2+5x}{h}=\dfrac{6xh+3h^2-5h}{h}=6x+3h-5.\]
As \(h\to 0\), this tends to \(6x-5\). So \(f'(x)=6x-5\) and, for instance, \(f'(2)=7\).

Example 3: a square root

Let \(f(x)=\sqrt{x}\) at \(a=4\). Multiply by the conjugate:
\[\dfrac{\sqrt{4+h}-2}{h}=\dfrac{(4+h)-4}{h\left(\sqrt{4+h}+2\right)}=\dfrac{1}{\sqrt{4+h}+2}\longrightarrow\dfrac{1}{4}.\]
So \(f'(4)=\dfrac14\): near \(x=4\), the output grows by about a quarter of the input change.

4. The tangent line equation

Tangent line

If \(f\) is differentiable at \(a\), the tangent line to the graph at \((a,f(a))\) has slope \(f'(a)\) and equation
\[y=f(a)+f'(a)\,(x-a).\]

Method: find a tangent line

Compute the point: \(f(a)\).
Compute the slope: \(f'(a)\).
Substitute into \(y=f(a)+f'(a)(x-a)\) and simplify.

Example 4: tangent line

Find the tangent to \(f(x)=x^3-3x\) at \(a=2\). We have \(f(2)=8-6=2\). Using the rules of Parts 6 and 7, \(f'(x)=3x^2-3\), so \(f'(2)=9\). The tangent is \(y=2+9(x-2)=9x-16\).

5. Differentiability and continuity

Theorem

If \(f\) is differentiable at \(a\), then \(f\) is continuous at \(a\).

Why. For \(h\neq 0\), write \(f(a+h)-f(a)=\dfrac{f(a+h)-f(a)}{h}\cdot h\). As \(h\to 0\) the first factor tends to \(f'(a)\) and the second to \(0\), so \(f(a+h)-f(a)\to 0\). That is exactly \(\lim_{x\to a}f(x)=f(a)\).

The converse is false

The function \(f(x)=|x|\) is continuous at \(0\), but not differentiable there. For \(h>0\) the quotient is \(\dfrac{|h|}{h}=1\); for \(h<0\) it is \(-1\). The two one-sided limits disagree, so \(f'(0)\) does not exist. A function typically fails to be differentiable at a corner, at a vertical tangent (the quotient tends to infinity), or at a discontinuity.

6. The power rule

Power rule

For any real number \(n\) (with \(x\) in the domain where the expressions make sense),
\[\dfrac{d}{dx}\left(x^n\right)=n\,x^{n-1}.\]
Special cases: \(\dfrac{d}{dx}(x)=1\) and, for any constant \(c\), \(\dfrac{d}{dx}(c)=0\).

For a positive integer \(n\), the binomial expansion gives \((x+h)^n=x^n+n x^{n-1}h+(\text{terms with }h^2)\), so the quotient is \(n x^{n-1}+(\text{terms with }h)\), which tends to \(n x^{n-1}\).

Example 5: rewrite, then apply the rule

\(\dfrac{1}{x^3}=x^{-3}\), so its derivative is \(-3x^{-4}=-\dfrac{3}{x^4}\). Also \(\sqrt{x}=x^{1/2}\), so its derivative is \(\dfrac12x^{-1/2}=\dfrac{1}{2\sqrt{x}}\), which matches Example 3: at \(x=4\) it gives \(\dfrac{1}{4}\).

Zyro’s tip

On my home planet we say: “Radical or fraction? Make it a power first.” Rewriting roots and reciprocals as powers of \(x\) lets you use one rule for everything.

7. Sum and constant multiple rules

Linearity of the derivative

If \(f\) and \(g\) are differentiable and \(c\) is a constant, then
\[(f+g)'=f'+g',\qquad (f-g)'=f'-g',\qquad (c\,f)'=c\,f'.\]

Example 6: a polynomial

For \(f(x)=4x^5-7x^3+2x-9\):
\[f'(x)=4\cdot 5x^4-7\cdot 3x^2+2\cdot 1-0=20x^4-21x^2+2.\]
Then \(f'(1)=20-21+2=1\).

Careful

The sum rule does not extend to products: \((f\cdot g)'\) is not \(f'\cdot g'\). For example \((x\cdot x)'=(x^2)'=2x\), not \(1\cdot 1\).

Key takeaways

The difference quotient \(\dfrac{f(a+h)-f(a)}{h}\) is the average rate of change and the slope of a secant line.
\(f'(a)=\lim_{h\to 0}\dfrac{f(a+h)-f(a)}{h}\) is the instantaneous rate of change and the slope of the tangent line.
Tangent line at \(a\): \(y=f(a)+f'(a)(x-a)\).
Differentiable at \(a\) implies continuous at \(a\); the converse fails (corners, vertical tangents).
Power rule: \((x^n)'=nx^{n-1}\); constants have derivative \(0\).
Sum and constant multiple rules let you differentiate polynomials term by term.