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Applied calculus topic summary: techniques of differentiation

Derivatives of Powers, Sums, and Constant Multiples
Power Rule
Sum and Constant Multiple Rules
In words: The derivative of a sum is the sum of the derivatives, and the derivative of a difference is the difference of the derivatives. The derivative of c times a function is c times the derivative of the function. 
Examples
Want some practice? Try the interactive tutorial or try some exercises. 

Marginal Analysis
If Q(x) represents any quantity such as cost, revenue, profit or loss on the sale of x items, then Q'(x) is called the marginal quantity. Thus, for instance, the marginal cost measures the increase in total cost per item. This is effectively the cost of each additional item. The marginal cost is distinct from the average cost, which measures the average of the total cost of the first x items. Average cost is given by

Examples
Suppose the cost of (the first) x items is given by C(x) = 4x^{0.2}  0.1x pounds Sterling.Then the marginal cost is C'(x) = 0.8x^{0.8}  0.1 pounds Sterling per unit.In particular, C'(3) ≈ 0.23 (pounds Sterling per unit) is the approximate cost of the third unit (or the fourth unit). The average cost of the first three units is


Product Rule
Product Rule In Words: Quotient Rule
Quotient Rule In Words: 
Examples
(The derivatives of f and g are shown in color.)
Quotient Rule
Of course, you should simplify the answers and not leave them like that! Press here for an online tutorial on the product & quotient rules. 

Combining Rules for Differentiation: Calculation Thought Experiment
The calculation thought experiment is a technique to determine whether to treat an algebraic expression as a product, quotient, sum, or difference. Given such an expression, consider the steps you would use in computing its value. If the last operation is multiplication, treat the expression as a product; if the last operation is division, treat the expression as a quotient, and so on. Using the Calculation Thought Experiment (CTE) to Differentiate a Function

Examples
1. (3x^{2} 4)(2x+1) can be computed by first calculating the expressions in parentheses and then multiplying. Since the last step is multiplication, we can treat the expression as a product. 2. (2x 1)/x can be computed by first calculating the numerator and denominator, and then dividing one by the other. Since the last step is division, we can treat the expression as a quotient. 3. x^{2} + (4x 1)(x+2) can be computed by first calculating x^{2}, then calculating the product (4x 1)(x+2), and finally adding the two answers. Thus, we can treat the expression as a sum. 4. (3x^{2} 1)^{5} can be computed by first calculating the expression in parentheses, and then raising the answer to the fifth power. Thus, we can treat the expression as a power. Using the CTE
Now we are left with two simpler functions to differentiate: x^{2}, which is a power, so we use the power rule, and (4x 1)(x+2), which is a product, so that we use the product rule on this:


Chain Rule
If f is a differentiable function of u and u is a differentiable function of x, then the composite f(u) is a differentiable function of x, and
Chain Rule In Words:
For instance, if f(u) = u^{0.5}, then
Generalized Differentiation Rules

Examples
(First look over the generlized rules on the left.) Press here for an online tutorial on the chain rule. 

Derivatives of Logarithmic and Exponential Functions
The following table summarizes the derivatives of logarithmic and exponential functions, as well as their chain rule counterparts (that is, the logarithmic and exponential functions of a function).

Examples
Press here for an online tutorial on derivatives of logarithmic and exponential functions. 

Derivatives of Trigonometric Functions
The following table summarizes the derivatives of the six trigonometric functions, as well as their chain rule counterparts (that is, the sine, cosine, etc. of a function).

Example
Press here for online text on derivatives of trigonometric functions. 

Implicit Functions and Implicit Differentiation
Given an equation in x and y, we can think of y as an implicit function of x. We can find dy/dx without first solving for y as follows:
Logarithmic differentiation

Example
Suppose we want to find dy/dx given that
