The sum and difference rule of derivatives of functions states that we can find the derivative by differentiating each term of the sum or difference separately. Therefore, we simply apply the power rule or any other applicable rule to differentiate each term in order to find the derivative of the entire function.

Here, we will learn how to find derivatives using the sum and difference rule. We will learn the formula that we can use, and we will apply it to solve some practice problems.

## Statement and formula for the sum and difference rule of derivatives

The sum and difference rule of derivatives states that the derivative of a sum or difference of functions is equal to the sum of the derivatives of each of the functions.

This means that when $latex y$ is made up of a sum or a difference of more than one function, we can find its derivative by differentiating each function individually.

The sum and difference rule of derivatives allows us to find the derivative of functions like the following:

$latex y=f(x)+g(x)$

In this case, its derivative is equal to:

$$\frac{dy}{dx}=f'(x) \pm g'(x)$$

This applies to the sum or difference of any number of functions.

To derive each of the functions or each of the terms, we use the power rule, $latex \frac{d}{dx}(x^n) = nx^{n-1}$, or any other applicable derivative rule.

## Steps to derive a sum or difference of two or more functions

Suppose we have to derive

$latex f(x) = x^2+5x$

We have a function that is a sum of two terms. Then, we can derive it by following these steps:

**Step 1:** Use the laws of exponents to transform radicals or rational expressions into exponential form. In this case, we have no radicals or rational expressions.

**Note:** An example would be to write $latex \sqrt{x}$ as $latex x^{\frac{1}{2}}$.

**Step 2:** Apply the power rule formula, $latex \frac{d}{dx}(x^n) = nx^{n-1}$, or other applicable rules to each term in the sum or difference:

$$f'(x) = 2x+5$$

**Step 3:** Simplify the resulting expression. In this case, we can no longer simplify.

**Note:** An example would be to write $latex x^{-\frac{1}{2}}$ as $latex \frac{1}{\sqrt{x}}$.

You can use $latex f'(x), y’,$ or $latex \frac{d}{dx}(f(x))$ as the derivative symbol on the left-hand side of the final answer.

## Examples of derivatives of a sum or difference of functions

Each of the following examples has its respective detailed solution, where we apply the power rule and the sum and difference rule.

**EXAMPLE 1**

Find the derivative of $latex f(x)=x^3+2x$.

##### Solution

**Step 1:** We don’t have radicals or variables written in rational form.

**Step 2:** We apply the power rule formula, $latex \frac{d}{dx}(x^n) = nx^{n-1}$, to derive both terms of the function:

$latex f(x)=x^3+2x$

$latex f'(x)=3x^2+2$

**Step 3:** The expression is now simplified.

**EXAMPLE **2

**EXAMPLE**What is the derivative of the function $latex f(x)=5x^4-5x^2$?

##### Solution

**Step 1:** The function has no radicals or rational expressions.

**Step 2:** Using the power rule, $latex \frac{d}{dx}(x^n) = nx^{n-1}$, we can differentiate both terms of the function:

$latex f(x)=5x^4-5x^2$

$latex f'(x)=20x^3-10x$

**Step 3:** The expression is now simplified.

**EXAMPLE **3

**EXAMPLE**Find the derivative of the function $latex f(x)=10x^7+5x^5+4x$.

##### Solution

**Step 1:** We have no radicals or rational expressions in the function.

**Step 2:** Applying the power rule, $latex \frac{d}{dx}(x^n) = nx^{n-1}$, to derive the three terms of the function, we have:

$latex f(x)=10x^7+5x^5+4x$

$latex f'(x)=70x^6+25x^4+4$

**Step 3:** The expression is now simplified.

**EXAMPLE **4

**EXAMPLE**Find the derivative of the function: $latex f(x) = -5x^{-3}+5x^2$.

##### Solution

**Step 1:** In this case, we have negative exponents, but we don’t have radicals or variables written in rational form.

**Step 2:** We apply the power rule formula, $latex \frac{d}{dx}(x^n) = nx^{n-1}$, to derive both terms of the function:

$latex f(x)=-5x^{-3}+5x^2$

$latex f'(x)=15x^{-4}+10x$

**Step 3:** We can use the laws of exponents to write the derivative as follows:

$$f'(x)=\frac{15}{x^4}+10x$$

**EXAMPLE **5

**EXAMPLE**What is the derivative of $latex f(x)=2x^4+4x^2+\sqrt{x}$?

##### Solution

**Step 1:** We have a square root in the function. Using the laws of exponents, we can write as follows:

$$f(x)=2x^4+4x^2+x^{\frac{1}{2}}$$

**Step 2:** Applying the power rule formula, $latex \frac{d}{dx}(x^n) = nx^{n-1}$, to differentiate the three terms of the function, we have:

$$f(x)=2x^4+4x^2+x^{\frac{1}{2}}$$

$$f'(x)=8x^3+8x+\frac{1}{2}x^{-\frac{1}{2}}$$

**Step 3:** Using the laws of exponents again, we can write as follows:

$$f'(x)=8x^3+8x+\frac{1}{2x^{\frac{1}{2}}}$$

**EXAMPLE **6

**EXAMPLE**Find the derivative of $latex f(x)=\frac{1}{\sqrt{x}}+\frac{3}{\sqrt[3]{x^2}}$.

##### Solution

**Step 1:** We use the laws of exponents to write in exponential form:

$$f(x)=x^{-\frac{1}{2}}+3x^{-\frac{2}{3}}$$

**Step 2:** Using the power rule, $latex \frac{d}{dx}(x^n) = nx^{n-1}$, we differentiate both terms of the function:

$$f(x)=x^{-\frac{1}{2}}+3x^{-\frac{2}{3}}$$

$$f'(x)=-\frac{1}{2}x^{-\frac{3}{2}}-2x^{-\frac{5}{3}}$$

**Step 3:** Using the laws of exponents again, we can write as follows:

$$f'(x)=-\frac{1}{2x^{\frac{3}{2}}}-\frac{2}{x^{\frac{5}{3}}}$$

$$f'(x)=-\frac{1}{2\sqrt{x^3}}-\frac{2}{\sqrt[3]{x^5}}$$

## Derivatives of sum and difference of functions – Practice problems

Solve the following practice problems using what you have learned about the sum and difference rule of derivatives, along with the power rule.

## See also

Interested in learning more about derivatives? You can take a look at these pages:

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