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2.2 Defining the Derivative of a Function and Using Derivative Notation

Syllabus
2020
Topic
2.2
Level
—

Learning objectives

—CHA-2.B—Represent the derivative of a function as the limit of a difference quotient—Topic 2.2• CHA-2.B Represent the derivative of a function as the limit of a difference quotient.• CHA-2.B.2 The derivative of f is the function whose value at x is lim h→0 (f(x + h) − f(x))/h, provided this limit exists.• CHA-2.B.3 For y = f(x), notations for the derivative include dy/dx, f′(x), and y′.• CHA-2.B.4 The derivative can be represented graphically, numerically, analytically, and verbally.• Enduring understanding CHA-2: Derivatives allow us to determine rates of change at an instant by applying limits to knowledge about rates of change over intervals.—CHA-2.C—Determine the equation of a line tangent to a curve at a given point• CHA-2.C Determine the equation of a line tangent to a curve at a given point.• CHA-2.C.1 The derivative of a function at a point is the slope of the line tangent to a graph of the function at that point.• Enduring understanding CHA-2: Derivatives allow us to determine rates of change at an instant by applying limits to knowledge about rates of change over intervals.

The Derivative as a Function and Across Representations

The derivative f′f' is a new function: at each input xx where the defining limit exists, f′(x)f'(x) gives the instantaneous rate of change of ff and the tangent slope at (x,f(x))(x,f(x)).

f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}

For y=f(x)y=f(x), the symbols f′(x)f'(x), y′y', and dydx\dfrac{dy}{dx} name the same derivative function. The expression f′(a)f'(a) is one value of that function: the rate and tangent slope specifically at x=ax=a.

If f(x)=x2f(x)=x^2, then (x+h)2−x2h=2x+h\dfrac{(x+h)^2-x^2}{h}=2x+h for h≠0h\ne0. Letting h→0h\to0 gives f′(x)=2xf'(x)=2x. Thus the derivative is a whole function; for example, f′(3)=6f'(3)=6 is one of its values.

Representation Meaning of f′(a)f'(a)
Graphical Slope of the tangent to ff at x=ax=a
Numerical Limiting value of nearby secant slopes
Analytical Value obtained from a derivative formula or defining limit
Verbal Instantaneous output change per input unit at aa

A derivative formula applies only at inputs where the defining limit exists. Do not confuse f′(x)f'(x), which varies with xx, with the single number f′(a)f'(a).

Writing the Tangent Line at a Point

At x=ax=a, the tangent line uses the point (a,f(a))(a,f(a)) on the curve and the slope f′(a)f'(a). Put those two pieces into point-slope form.

y-f(a)=f'(a)(x-a)

  1. Evaluate f(a)f(a) to find the point on the curve.
  2. Evaluate f′(a)f'(a) to find the tangent slope.
  3. Substitute both values into point-slope form.
  4. Simplify only if another line form is requested.

For f(x)=x2f(x)=x^2 at x=3x=3, the curve point is (3,9)(3,9) and f′(x)=2xf'(x)=2x gives slope f′(3)=6f'(3)=6. Therefore y−9=6(x−3)y-9=6(x-3), or equivalently y=6x−9y=6x-9. Substituting x=3x=3 returns y=9y=9, confirming that the line passes through the required point.

Use f(a)f(a) as the point's yy-coordinate and f′(a)f'(a) only as the slope; interchanging them gives the wrong line. A tangent line describes the curve's local linear behavior and may cross the curve elsewhere.