Gerald has taught engineering, math and science and has a doctorate in electrical engineering.
Calculus: A Baking Analogy
It's your first day in the kitchen, and you want to bake a cake. What if the recipe simply states to 'just add butter'? Your response would probably be: 'Hey, not enough information! We need formulas, like three teaspoons equal one tablespoon. We need rules like, bake until cooked.'
Although calculus is usually not used to bake a cake, it does have both rules and formulas that can help you figure out the areas underneath complex functions on a graph. In this lesson we will focus on the formulas and rules for both differentiation, the method by which we calculate the derivative of a function, and integration, the process by which we calculate the antiderivative of a function.
Looking at Differentiation
Before we dive into formulas and rules for differentiation, let's look at some notation for differentiation. We can write the derivative of a function f(x) as:
We read these as: d by dx of f of x, f' prime of f of x, df of x and cap Df of x.
Although these and other notations are used for differentiation, we will use the d by dx and the prime notations in this lesson.
Here are some differentiation formulas:
You may have noticed in the first differentiation formula that there is an underlying rule. It's called the power rule, which says that the derivative of x raised to the power n is n times x raised to the power n minus 1.
This rule works even when n is not an integer. For example, the square root of x is x raised to the ½ power. Using the power rule for differentiation, we see how this works out in the example equation here:
As an example, let's show that the derivative of a constant is consistent with the power rule by noting that x^0 is equal to one. When we differentiate the constant C, we are in fact differentiating Cx^0. What you're looking at below is what we wanted to show:
In this last example, we used the coefficient rule, which states that the derivative of a constant times a function is that constant times the derivative of the function.
Here are some other useful rules for differentiation, such as the product, quotient and chain rules. The product rule states that the derivative of the product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. As an example, check out the product rule in the action now:
There's also a way to find the derivative of one function divided by another function, the quotient rule, which says that the derivative of the quotient of two functions is the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator all divided by the denominator squared. You can see this play out in the example below:
Instead of simply having x as the argument of a function, the argument might be another function. In these cases, we use the chain rule, which states that when we have a function g(x) within a function f, the resulting derivative is the derivative of the function f multiplied by the derivative of the function g. You can see this play out in this last example:
Looking at Integration
The notation for integration takes the form of:
We read this as: the integral of f of x times dx, which is called an indefinite integral. On the other hand, a definite integral will have a lower limit and an upper limit that we write as:
We read this as: the integral from x1 to x2 of f of x times dx.
Here are some integration formulas for definite integrals:
Now, let's consider some rules for integration. The coefficient rule states that the integral of a constant times a function is that constant times the integral of the function, as shown below:
Just as with differentiation, there is a power rule for integration. The integral of x raised to a power n is x^(n+1) divided by (n+1) plus a constant C. You can see this in our example below:
The substitution rule allows us to simplify an integral when we can determine a variable u that can be substituted for a function in the integral.
Let's look at another example and find the following answers you can see here:
The integration by parts rule states that the integral of u dv is uv minus the integral of v du.
For example, letting u = x and dv = e^x * dx, we get du = (1)dx = dx and v = e^x. From there, we get the solution you can see here:
In basic calculus, we learn rules and formulas for differentiation, which is the method by which we calculate the derivative of a function, and integration, which is the process by which we calculate the antiderivative of a function. These formulas allow us to deal with powers of x, constants, exponentials, natural logarithms, sines and cosines. The rules of differentiation include the following:
- Constant times a function rule for differentiation
- Power rule for differentiation
- Product rule for differentiation
- Quotient rule for differentiation
- Chain rule for differentiation of nested functions
The rules of integration include the following:
- Constant times a function rule for integration
- Power rule for integration
- Substitution rule for integration
- Integration by parts rule
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