Basic Rules of Differentiation Study Help
Basic Rules of Differentiation
Using the limit definition to find derivatives can be very tedious. Luckily, there are many shortcuts available. For example, if function f is a constant, like f (x) = 5 or f (x) = 18, then f' (x) = 0. This can be proven for all constants c at the same time in the following manner.
f (x) = c
All of the general rules in this chapter can be proven in such a manner, using the limit definition of the derivative, though we shall not bother to do so. The first rule is the Constant Rule, which says that if f (x) = c where c is a constant, then f' (x) = 0.
Before we go any further, a word needs to be said about notation. The concept of the derivative was discovered by both Isaac Newton and Gottfried Leibniz. Newton would put a dot over an object to represent its derivative, much like the way f' (x) represents the derivative of f (x). Leibniz would write the derivative of y (where x is the variable) as . Newton's notation is certainly more convenient, but Leibniz's enables us to represent "take the derivative of something" as (something). Thus, if y = f (x) , then = (f (x)) = f' (x).Using Leibniz's notation, the Constant Rule where c is a constant is (c) = 0.
We will use both forms of the Constant Rule, depending on the situation. The next rule is the Power Rule, which is stated: (xn) = n · xn – 1. This rule says "multiply the exponent in front and then subtract one from it."
If f (x) = c where c is a constant, then f'(x) = 0.
And, using Leibnez's notation, if c is a constant, then (c) = 0.
Differentiate f(x) = x2.
f' (x) = 2x2–1 = 2x1 = 2x
Differentiate y = x8.
Differentiate g(x) = √x.
To use the Power Rule, we need g(x) expressed as x raised to a power, or:
Notice how much easier it is to use the Power Rule to solve this problem than it was using the limit definition of the derivative.
Differentiate y =
Again, we have to rewrite y as x–2 so that it becomes x raised to a power.
Differentiate y = 3√t.
Notice that means «take the derivative with respect to variable t." Usually, our variable is x, so the derivative of , but sometimes, we have other variables. If y = f(u), then is the derivative with respect to u, for example.
The Constant Coefficient Rule
The Constant Coefficient Rule is stated as follows: If a function has a constant multiplied in front, leave it while you take the derivative of the rest. This means that because then the derivative of 5x8 is 5 · (8x7) = 40x7. Just imagine that the constant steps aside and waits while you differentiate the rest.
The Constant Coefficient Rule
If a function has a constant multiplied in front, leave it while you take the derivative of the rest.
Differentiate the following.
|f(x) = 11x4|
|y = 10x2|
|g(x) = 3√x = 3x|
|h(t) = = 4t–6|
|y = 12x|
|k(u) = =|
|A(r) = π r2|
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