D - Differentiation Rules Lesson

Differentiation Rules

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

 $\frac{d}{dx}\left(c\right)=0$

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

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

For the function , 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

 $\frac{d}{dx}\left[cf\left(x\right)\right]=c\frac{d}{dx}f\left(x\right)$

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. $\frac{d}{dx}cf\left(x\right)=\lim_{h \to 0}\frac{cf\left(x+h\right)-cf\left(x\right)}{h}=c\lim_{h \to 0}\frac{f\left(x+h\right)-f\left(x\right)}{h}=c\frac{d}{dx}f\left(x\right)$

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

Power Rule

If n is any real number, then $\frac{d}{dx}\left(x^n\right)=nx^{n-1}$, for all x where the powers xⁿ and x^n-1 are defined. View a proof of this rule below.

For the function f(x) = x³³, the derivative f '(x) = 33 x³². 

Sum and Different Rules

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

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

Given $y=x^3-5x^2+4x$ the derivative is $y=3x^2-10x+4$.

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

The Product Rule

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

 $\frac{d}{dx}\left[f\left(x\right)g\left(x\right)\right]=f\left(x\right)\frac{d}{dx}g\left(x\right)\frac{d}{dx}g\left[f\left(x\right)\right]$

In words, the Product Rule says that the derivative of a product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function.

Rollover the following function to see what occurs when the Product Rule is applied. $h\left(x\right)=\left(2x+3\right)\left(1-x\right)$

View the presentation below on applying the product rule and finding the equation of a tangent line to a curve defined by the product of two functions using the TI-84.

The Quotient Rule

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

$\frac{d}{dx}\left[\frac{f\left(x\right)}{g\left(x\right)}\right]=\frac{g\left(x\right)\frac{d}{dx}\left[f\left(x\right)\right]-f\left(x\right)\frac{d}{dx}\left[g\left(x\right)\right]}{\left[g\left(x\right)\right]^2}$

In words, the Quotient Rules says that the derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. A much less formal interpretation is "low d high minus high d low over low squared."

Rollover the following function to see what occurs when the Quotient Rule is applied. $h\left(x\right)=\frac{2x+3}{1-x}$.

Click HERE to participate in several self-check opportunities. Links to an external site.  Simply enter the words "derivative of" followed by the expression, e.g., derivative of (x^3 + x - 1)/(x + 3). 

Differentiation Rules Practice

Differentiation Rules: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of differentiation rules.

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