LINEARITY OF THE DERIVATIVE
Linear Combinations of Functions
We can build new functions from old ones by forming sums and constant multiples.
Given functions f and g with target sets consisting of real numbers, the sum of f and g is the new function f + g defined as follows.
(f + g)(x) = f(x) + g(x)
The domain of f + g is the set of all points x that lie both in the domain of f and in the domain of g.
Given a function f with target set consisting of real numbers and a real number c, we can form the constant multiple cf, which is the new function defined in this way.
(cf)(x) = cf(x)
The domain of cf is the same as the domain of f. Using sums and constant multiples of functions, we are able to subtract functions as well.
Example. Given a function f, the additive inverse -f is the constant multiple (-1)f.
Example. Given f and g, the difference of f and g is given by f - g = f + (-1)g
If we are given a bunch of functions f, g, h,..., then any function that can be built by forming sums and constant multiples of f, g, h,... is called a linear combination of these functions.
Example. A linear combination of the identity function (y = x) and a constant function gives rise to a function whose graph is a nonvertical horizontal line: y = mx + b. The slope of this graph is m and the y-axis intercept is the point (0,b).
A linear combination of differentiable functions is differentiable. Our first general differentiation rule shows us how to differentiate a linear combination of functions.
Linearity of the Derivative
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(f(x) + g(x)) = f '(x) + g'(x) |
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Why is THAT true?
Here's what these two formulas say in words, together with a third statement that is an immediate consequence of the first two.
- The derivative of a sum is the sum of the derivatives.
- The derivative of a constant multiple is the constant multiple of the derivative.
- The derivative of a linear combination is the linear combination of the derivatives.
Example. In this example, we differentiate a linear combination of the sine and square root functions. Recall that Dx(sin(x)) = cos(x) and that Dx(sqrt(x)) = 1/2sqrt(x).
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You can inspect the graphs of f(x) = 3sin(x) - 5sqrt(x) and its derivative by pushing the button.
Example. The quadratic function q(x) = 5x2 - 4x - 3 is a linear combination of the squaring function (y = x2), the identity function (y = x), and a constant function (y = 1). Knowing that Dx(x2) = 2x, we can easily differentiate the function q.
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5(2x) - 4(1) + 0 = 10x - 4 |
For a very little work, we learn that q'(x) = 10x - 4. Note that the derivative of the quadratic function q is a linear function.
We continue with a differentiation rule for products of functions.
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© CalculusQuestTM
Version 1
All rights reserved---1996
William A. Bogley
Robby Robson