Derivative: Definition
How to find the derivative of a function at the point on the line by using the definition of the derivative: definition, 1 example, and its solution.
Definition
Think of two points on the graph y = f(x):
(a, f(a)) and (a + h, f(a + h)).
Draw a line that passes through these two points.
The slope of the line is
[f(a + h) - f(a)]/[(a + h) - a]
= [f(a + h) - f(a)]/h.
The change of x is h.
Then, as h → 0,
(a + h, f(a + h)) goes to (a, f(a)).
Then the line becomes the tangent of y = f(x) at x = a.
Limit of a Sequence
f'(a), f prime a, is the slope of y = f(x) at x = a.
So f'(a) = limh → 0 [f(a + h) - f(a)]/h.
This is the definition of the derivative f'(a).
Examplef(x) = x2 - 3x + 1, f'(2) = ?
f'(2) = limh → 0 [f(2 + h) - f(2)]/h.
f(x) = x2 - 3x + 1
Then f(2 + h) = (2 + h)2 - 3(2 + h) + 1.
And f(2) = 22 - 3⋅2 + 1.
(2 + h)2 = 22 + 2⋅2⋅h + h2 = 4 + 4h + h2
Square of a Sum
-3(2 + h) = -6 - 3h
Multiply a Monomial and a Polynomial
-[22 - 3⋅2 + 1] = -[4 - 6 + 1] = -4 + 6 - 1
Cancel 4 and -4.
Cancel -6 and +6.
And cancel +1 and -1.
+4h - 3h = +h
Then limh → 0 [h2 + h]/h.
Divide both of the numerator and the denominator by h.
Then limh → 0 [h + 1]/1.
Then limh → 0 [h + 1]/1 = [0 + 1]/1 = 1.
So f'(2) = 1.
This curve is the graph of y = f(x):
y = x2 - 3x + 1.
f'(2) = 1 means
the slope of y = f(x) at x = 2 is 1.