Differentiation is easy to deal with the following rules. It is considered to be easier than solving integrals. Most of the integrals can be evaluated using an antiderivative calculator while derivatives have some rules that students should grasp on priority.
These rules should be applied to according to the equation or underlying problem. We will explain the basic rules for students so that they can solve derivative problems like a pro.
1. Power Rule
The power rule is applied to polynomial functions. The formula is:
The example of f(x2) we solved to learn derivatives is a perfect representation of the power rule. But for more understanding, see another example:
Example:
Derivative of x6. Solve this using limits as well as power rules.
Solution:
Solving using limits.
Step 1: Add delta x i.e and expand the equation.
= (x + )6
f(x + )6 = (x6 + 6x5
+ … +
6)
Step 2: Apply the formula of the slope.
=
=
= … cancelling x6
=
= 6x5 + 15x4 x + … +
x5
Step 3: Apply limit.
f’(x6) =6x5 + 15x4
x + … +
x5
= 6x5
Solving using power rule.
Step 1: Write the function.
f(x) = x6
Step 2: Identify the components and apply the power rule.
As yn = nyn-1, (we formed the formula with y to remove any confusion.)
Here, y = x
n = 6
Applying the rule;
f’(x6) = (6)x6-1 = 6x5
The results of both methods are the same. This proves that the power rule is true for any n.
2. Sum & Difference Rule
Two functions in addition or difference are also eligible for a differentiation rule. The rules applied in such situations are;
f + g = f +
g
f - g = f -
g
Solution:
Step 1: Apply derivative.
= ( 2x + 3x2)
Step 2: Apply the rule.
= 2x +
3x2
= 2.1x1-1 + 3.2x2-1 applying the power rule.
= 2 + 6x
3. Product Rule
When the derivative of two functions in multiplications is computed, we then use the product rule. An example of such a function will be 4x4(3x + 9).
The formula of product rule is:
f(x)g(x) = f
g + g
f
What is the derivative of 4x4(3x + 9)?
Solution:
Step 1: Apply derivative.
=4x4(3x + 9)
According to the product rule.
=4x4(3x + 9) + (3x + 9)
4x4
Step 2: Solve for (3x + 9).
=(3x + 9)
Apply sum rule.
=3x +
9
= 3 + 0 … (power rule for 3x and constant rule for 9)
= 3
Step 3: Solve for 4x4.
Using power rule;
= 4.{(4)x4-1}
= 4.4x3
= 16x3
Step 4: Put the values and simplify.
= 4x4.3 + (3x +9)16x3
= 12x4 + 48x4 + 144x3
= 60x4 + 144x3
= x3(60x + 144)
4. Constant Rule
It is probably the simplest derivative rule. When we don’t have a variable in a function e.g y=4, then the derivative is 0.
f’(c) = 0
Where c is a constant number. This rule makes sense if you try to visualize it.
Since x = 0, hence there is no slope. And the graph is a straight vertical line. But let’s see this with the help of differentiation.
Example:
What is the derivative of f(x) = 5.
Solution:
Step 1: Add delta x.
f(x + ) = 3 +
= 3
Step 2: Apply formula.
=
=
= 0/ = 0
5. Quotient Rule
We have a formula for product but do we have a formula for division? Yes! And It is called the quotient rule. It is mainly derived from product rule for differentiation.
=
A quotient equation looks something like this: . To find its derivative, it is divided into two parts;
. You can see that actually, we have to perform the product rule.
= f(x)
+
f
All we need to do is to find the derivative of 1/g(x). Following all the familiar process of applying formula and limit, we will get:
Note that is the
with a negative sign.
Or it can be written as:
Now on putting this value in the product formula we discussed above:
=
+
Taking LCM.
=
=
This is the quotient rule.
Find the derivative of (x - 3)/(x3 + x).
Solution:
Directly apply the quotient rule.
Step 1: Find the derivative of (x-3).
=(x-3)
= 1
Step 2: Find derivative of (x3-x)
=(x3-x)
Using the rules discussed above;
= 3x2 - 1
Step 3: Use these in the main equation.
After simplifying;
6. Chain rule
Last but not the least, we have the chain rule. It states that:
or f(g(x)) = f’{g(x)}g’(x)
This is for simplicity's sake. When a function has many mathematical operations, it is better to convert it in terms of two functions e.g, we have a function:
f(x)= (X2 + 6)4
Let g =X2 + 6, then
f(g) = (g)4 where g =X2 + 6
f(g(x)) = (X2 + 6)4
To find its derivative, you first find the difference of f with respect to g and then g’(x). After that, multiply.
Solution:
Let g(x) = x2 + 6
And f(g) = g4
So the derivative of f(g(x)) = f(g(x)) =f’(g(x)).g’(x).
Applying the chain rule.
Step 1: Find the derivative of g(x).
= x2 + 6 … using power rule for x2 and constant rule for 6
= 2x
Step 2: Find the derivative of f(g).
= g4
= 4g3
Step 3: Putting in the formula and multiplying.
= 4(x2+6)3. 2x
= 8x.(x2+6)3
Wrapping up
All of the above listed rules of derivatives are most common rules that students have to deal with during their academic course. You should practice these rules to avoid any step backs during your exams. You should also try differentiation calculator after practicing these rules. This tool will help you out in solving derivatives while performing differentiation.
