Derivative Rules: Constant, Sum, and Difference Lesson

Constant Rule and Constant Multiple Rule

The constant function f(x) = c is the simplest of all functions. Its graph is the horizontal line y = c with a slope of 0. Thus, f '(x) must equal 0. The derivative of a constant function is given by

equation image indicator $LaTeX: \frac{d}{dx}\left(c\right)=0$

To prove this rule, we apply the definition of the derivative.

equation image indicator $LaTeX: f'\left(x\right)=\lim_{h \to 0}\frac{f\left(x+h\right)-f\left(x\right)}{h}=\lim_{h \to 0}\frac{c-c}{h}=0$

For the function equation image indicator , the derivative f '(x) = 0.

If f is a differentiable function of x and c is a constant, then the derivative of a constant multiple of a function is given by

equation image indicator $LaTeX: \frac{d}{dx}\left[cf\left(x\right)\right]=c\frac{d}{dx}f\left(x\right)$

In words, the derivative of a constant times a function is equal to the constant times the derivative of the function. As before, proof of this rule uses the definition of a derivative. $LaTeX: \frac{d}{dx}cf\left(x\right)=\lim_{h \to 0}\frac{cf\left(x+h\right)-cf\left(x\right)}{h}=c\lim_{h \to 0}\frac{f\left(x+h\right)-f\left(x\right)}{h}=c\frac{d}{dx}f\left(x\right)$

For the function f(x) = 4x, the derivative is f '(x) = 4.

Sum and Different Rules

If f and g are both differentiable functions of x, then

equation image indicator $LaTeX: \frac{d}{dx}\left[f\left(x\right)+g\left(x\right)\right]=\frac{d}{dx}f\left(x\right)+\frac{d}{dx}g\left(x\right)\\ and\\ \frac{d}{dx}\left[f\left(x\right)-g\left(x\right)\right]=\frac{d}{dx}f\left(x\right)-\frac{d}{dx}g\left(x\right)$

Given equation image indicator LaTeX: y=x^3-5x^2+4x the derivative is LaTeX: y=3x^2-10x+4 .

Both the Sum Rule and the Difference Rule can be extended to the sum or difference of any number of functions.

Differentiation Rules: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of the constant, constant multiple, sum and difference differentiation rules.

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