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Sum Rule – Explanation and Examples

 

What is Sum Rule?

We know how to take the derivatives of single expressions. But what happens when there are two expressions, three expressions, or even more? How do we take the derivative?

We use the sum rule. The take each individual expressions’ derivative, then, take its sum. Very simple! This rule also applies for limits. If we have to evaluate a limit consisting of several terms, we can evaluate each individual terms’ limits and then sum it.

In calculus, the sum rule is actually a set of 3 rules. We have the sum rule for limits, derivatives, and integration. We use the sum rule when we have a function that is a sum of other smaller functions.

At this point, we will look at sum rule of limits and sum rule of derivatives.

Let’s take a look at its definition.

Definition of Sum Rule of Limits: The sum rule of limits tells us that the limit of a sum is the sum of the limits.

Definition of Sum Rule of Derivatives: The sum rule of derivative tells us that the derivative of a sum is the sum of the derivatives.

– Sum Rule of Limits

Definition: The limit of a sum is the sum of the limits. To take the limit of a sum, you basically take the individual limits of each expression and add them. Suppose we have a function h(x) that can be broken apart as h(x)=f(x)+g(x).

The mathematical definition would be:

lim┬(x→a)⁡h(x)
=lim┬(x→a)⁡(f(x)+g(x))
=lim┬(x→a)⁡〖f(x)〗+lim┬(x→a)⁡〖g(x)〗

Let’s look at a couple of examples:

Example 1:

Find the limit of the function:

lim┬(x→3)⁡〖〖(x〗^3+4)〗

Solution:

We have a sum:

(x^3+4)

We can break it apart into x^3 and 5. We take the individual limits and evaluate:

lim┬(x→3)⁡(x^3+4)
=lim┬(x→3)⁡〖x^3 〗+lim┬(x→3)⁡4
=(3)^3+4
=27+4
=31

Note: The limit of a constant function a is always a no matter what the x value approaches.

Example 2:

Find the limit of the function:

lim┬(x→1)⁡〖(x^4+12x)〗

Solution:

We have a sum:

(x^4+12x)

We can break it apart into x^4 and 12x. We take the individual limits and evaluate:

lim┬(x→1)⁡(x^4+12x)
=lim┬(x→1)⁡〖x^4 〗+lim┬(x→1)⁡12x
=(1)^4+12(1)
=1+12
=13

– Extended Sum Rule of Limits

The extended sum rule of limits tells us that if we have a sum of n expressions, the limit of that expression would be the sum of each of the limits.

Mathematically:

lim┬(x→a)⁡(f_1 (x)+f_2 (x)+⋯+f_n (x))=lim┬(x→a)⁡〖f_1 (x)〗+lim┬(x→a)⁡〖f_2 (x)〗+⋯+lim┬(x→a)⁡〖f_n (x)〗

So, no matter how many expressions add up to the whole expression of which we are taking the limit, we can simply find each individual expressions’ limit and sum all of them.

Let’s look at another example:

Example 3:

Find the limit of the function:

lim┬(x→1)⁡〖〖(x〗^4+x^2+2x+2)〗

Solution:

We break apart the expression into individual expressions and take each limit separately. Then we sum it up to get our final answer.

Shown below:

lim┬(x→1)⁡〖〖(x〗^4+x^2+2x+2)〗
=lim┬(x→1)⁡〖x^4 〗+lim┬(x→1)⁡〖x^2 〗+lim┬(x→1)⁡2x+lim┬(x→1)⁡2
=(1)^4+(1)^2+2(1)+2
=1+1+2+2
=6

Note: Remember, the limit of a constant function a is always a no matter what the x value approaches

– Sum Rule of Derivatives

Definition: The derivative of a sum is equal to the sum of the derivatives. To take the derivative of a sum, you break apart the sum into single expressions, take the derivatives, and add them. Suppose we have a sum f(x)+g(x). To take its derivative, we take the derivative of f(x) and g(x) individually, and them take the sum.

Mathematically:

d/dx [f(x)+g(x)]
=d/dx [f(x)]+d/dx [g(x)]
=f^’ (x)+g'(x)

Note: f^’ (x) is the notation for the derivative of the function f(x).

Let’s look at an example to clarify:

Example 4:

Find the derivative of the function:

h(x)=2x^2+3x

Solution:

We break apart h(x) into 2x^2 and 3x. Then we take the individual derivatives and sum them. Shown below:

d/dx [h(x)]
=d/dx (2x^2 )+d/dx (3x)
=4x+3

Note: We used the sum rule of derivatives to break it apart. We also used the power rule to do the actual differentiation.

– Proof of Sum Rule of Derivatives

To prove the sum rule of derivatives, we recall the definition of a derivative.

Definition: The derivative of f(x) with respect to x is the function f'(x) which is defined as:

f^’ (x)=lim┬(h→0)⁡〖(f(x+h)-f(x))/h〗

Now, let us prove the sum rule of derivatives. Let h be a function that is sum of two functions, f and g.

h(x)=f(x)+g(x)

By using the definition of a derivative, we can write the function h(x) as:

h^’ (x)=lim┬(h→0)⁡〖(f(x+h)+g(x+h)-[f(x)+g(x)])/h〗

We re arrange this as:

h^’ (x)=lim┬(h→0)⁡〖(f(x+h)-f(x)+g(x+h)-g(x))/h〗

Further re arranging gives us:

h^’ (x)=lim┬(h→0)⁡((f(x+h)-f(x))/h+(g(x+h)-g(x))/h)

Using the sum rule of limits, we can write this as:

h^’ (x)=lim┬(h→0)⁡〖(f(x+h)-f(x))/h〗+lim┬(h→0)⁡〖(g(x+h)-g(x))/h〗

Note: Recall that the sum rule of limits can be used to break apart an expression into a sum and take individual limits of each broken down expression and add them to get the limit of the whole expression.

We recognize that lim┬(h→0)⁡〖(f(x+h)-f(x))/h〗 is the derivative of f(x), or f'(x) and lim┬(h→0)⁡〖(g(x+h)-g(x))/h〗 is the derivative of g(x), or g'(x).

Thus, we can now write it as:

h^’ (x)=f^’ (x)+g'(x)

So, we clearly see that the derivative of a sum is equal to the sum of the derivatives.

– Extended Sum Rule of Derivatives

The extended sum rule of derivative tells us that if we have a sum of n functions, the derivative of that function would be the sum of each of the individual derivatives. Mathematically:

d/dx [f_1 (x)+⋯+f_n (x)]=d/dx [f_1 (x)]+⋯+d/dx[f_n (x)]

So, no matter how many expressions add up to the whole expression of which we are taking the derivative, we can simply find each individual expressions’ derivative and sum all of them.

Let’s look at another example:

Example 5:

Find the derivative of the function shown below:

f(x)=3x+x^4+2x^2+10

Solution:

We break apart the function into individual expressions and take each derivative separately. Then we sum it up to get our final answer. Shown below:

d/dx [f(x)]
=d/dx (3x)+d/dx (x^4 )+d/dx (2x^2 )+d/dx (10)
=3+4x^3+4x+0
=4x^3+4x+3

Note: Remember, the derivative of a constant function a is always 0. Here, we used the power rule of derivatives to do the differentiation.

Sometimes we have to use the sum rule of derivatives on functions that are in disguise!

Let’s look at an example:

Example 6:

Find the derivative of the function shown below:

f(x)=(2x^4 √x+3x)/√x

Solution:

We have to rewrite the function f(x) into a sum by simplifying. The process is shown below:

(2x^4 √x+3x)/√x
=(2x^4 √x)/√x+3x/√x
=2x^4+3x^(1/2)

Now, it is a sum. Let’s use the sum rule of derivatives on this to figure out the answer:

d/dx [f(x)]
=d/dx [(2x^4 √x+3x)/√x]
=d/dx [2x^4+3x^(1/2) ]
=d/dx (2x^4 )+d/dx (3x^(1/2) )
=8x^3+3/2 x^(-1/2)

Note: Again, we used the power rule to do the differentiation.

Summary

Practice Problems

1. Compute the limit shown below using the sum rule of limits:

lim┬(x→-1)⁡〖(2x^2+x)〗

2. Compute the limit shown below using the sum rule of limits:

lim┬(x→1)⁡〖(x^5+2x^3-10x+5)〗

3. Compute the limit shown below using the sum rule of limits:

lim┬(x→2)⁡〖(-3x^(3 )+4x+x^2-13)〗

4. Compute the limit shown below using the sum rule of limits:

lim┬(x→t)⁡(x^4+x^3+x^2+x+1)

5. What is the limit of lim┬(x→-2)⁡〖(3x+4+x^2)〗 ?

a) 14
b) 2
c) 10
d) -6

6. Find the derivative of f(x):

f(x)=4x+5x^2

7. Find the derivative of h(x):

h(x)=1/2 (1/3 x^2+x^3 )

8. Find the derivative of g(x):

g(x)=1/√x+√x

9. Find the derivative of p(x):

p(x)=(x+3)(x+1)

10. What is the derivative of f(x)=-2x+3x^2-3

a) f^’ (x)=-2+3x-3
b) f^’ (x)=-2+6x-3
c) f^’ (x)=-2+6x
d) f^’ (x)=-2+9x