To differentiate a sum or difference of functions, we have to differentiate each term of the function separately. This means that we can simply apply the power rule or another relevant rule to differentiate each term in order to find the derivative of the entire function.

Here, we will solve 10 examples of derivatives of sum and difference of functions. In addition, we will explore 5 problems to practice the application of the sum and difference rule.

##### CALCULUS

**Relevant for**…

Solving examples of derivatives of sum and difference of functions.

##### CALCULUS

**Relevant for**…

Solving examples of derivatives of sum and difference of functions.

## 10 Examples of derivatives of sum and difference of functions

The following examples have a detailed solution, where we apply the power rule, and the sum and difference rule to derive the functions.

**EXAMPLE 1**

Find the derivative of $latex f(x)=x^4+5x$.

##### Solution

**Step 1:** We start by converting radical or rational expressions to their exponential form. In this case, we don’t have radicals or variables written in rational form.

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

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

$latex f'(x)=4(x^{4-1})+5(x^{1-1})$

$latex f'(x)=4x^3+5$

**Step 3:** We simplify if possible. In this case, the expression is simplified.

**EXAMPLE **2

**EXAMPLE**

What is the derivative of the function $latex f(x)=-5x^3+10x^2$?

##### Solution

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

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

$latex f(x)=-5x^3+10x^2$

$latex f(x)=3(-5x^{3-1})+2(10x^{2-1})$

$latex f'(x)=-15x^2+20x$

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

**EXAMPLE **3

**EXAMPLE**

Find the derivative of the function $latex f(x)=7x^8+5x^{-3}$.

##### Solution

**Step 1:** Both terms have variables with numerical exponents.

**Step 2:** We have a negative exponent, but we simply use the power rule to differentiate both terms:

$latex f(x)=7x^8+5x^{-3}$

$latex f(x)=8(7x^{8-1})+(-3)(5x^{-3-1})$

$latex f'(x)=56x^7-15x^{-4}$

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

$$f'(x)=56x^7-\frac{15}{x^4}$$

**EXAMPLE **4

**EXAMPLE**

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

##### Solution

**Step 1:** We have negative exponents, but there are no radicals or variables with rational expressions.

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

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

$latex f(x)=-5(3x^{-5-1})-2(-2x^{-2-1})$

$latex f'(x)=-15x^{-6}+4x^{-3}$

**Step 3:** We use the laws of exponents to simplify:

$$f'(x)=-\frac{15}{x^6}+\frac{4}{x^3}$$

**EXAMPLE **5

**EXAMPLE**

What is the derivative of the function $latex f(x)=-5x^4+ \frac{1}{x}$?

##### Solution

**Step 1:** Using the laws of exponents, we can write as follows:

$latex f(x)=-5x^4+ x^{-1}$

**Step 2:** Now that we only have numerical exponents, we use the power rule formula, $latex \frac{d}{dx}(x^n) = nx^{n-1}$, on both terms of the function:

$latex f(x)=-5x^4+ x^{-1}$

$latex f'(x)=-20x^3-x^{-2}$

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

$$f'(x)=-20x^3-\frac{1}{x^2}$$

**EXAMPLE **6

**EXAMPLE**

What is the derivative of $latex f(x)=4x^3+2x^2+\frac{2}{x^3}$?

##### Solution

**Step 1:** We start by converting the rational expression to an expression with a numerical exponent:

$latex f(x)=4x^3+2x^2+2x^{-3}$

**Step 2:** Using the power rule for the three terms, we have:

$latex f(x)=4x^3+2x^2+2x^{-3}$

$latex f'(x)=12x^2+4x-6x^{-4}$

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

$$f'(x)=12x^2+4x-\frac{6}{x^4}$$

**EXAMPLE **7

**EXAMPLE**

Determine the derivative of $latex f(x)=3x^2+\frac{2}{3x^2}+ x^{ \frac{1}{2}}$.

##### Solution

**Step 1:** We use the laws of exponents to write the function like this:

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

**Step 2:** Using the power rule on the three terms of the function, we have:

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

$$f'(x)=6x-\frac{4}{3}x^{-3}-\frac{1}{2}x^{-\frac{3}{2}}$$

**Step 3:** We simplify by writing as follows:

$$f'(x)=6x-\frac{4}{3x^3}-\frac{1}{2x^{\frac{3}{2}}}$$

$$f'(x)=6x-\frac{4}{3x^3}-\frac{1}{2\sqrt{x^3}}$$

**EXAMPLE **8

**EXAMPLE**

Find the derivative of the function $latex f(x)=\sqrt{x}+ \frac{3}{x^3} $.

##### Solution

**Step 1:** Applying the laws of exponents, we can write as follows:

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

**Step 2:** We apply the power rule to derive both terms:

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

$$f'(x)=\frac{1}{2}x^{-\frac{1}{2}}-9x^{-4}$$

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

$$f'(x)=\frac{1}{2x^{\frac{1}{2}}}-\frac{9}{x^4}$$

$$f'(x)=\frac{1}{2\sqrt{x}}-\frac{9}{x^4}$$

**EXAMPLE **9

**EXAMPLE**

Find the derivative of $latex f(x)=5x^{-5}+\frac{1}{\sqrt{x}}$.

##### Solution

**Step 1:** We start by writing as follows:

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

**Step 2:** Differentiating both terms with the power rule, we have:

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

$$f'(x)=-25x^{-6}-\frac{1}{2}x^{-\frac{3}{2}}$$

**Step 3:** We simplify the resulting expression using the laws of exponents:

$$f'(x)=-\frac{25}{x^6}-\frac{1}{2x^{\frac{3}{2}}}$$

$$f'(x)=-\frac{25}{x^6}-\frac{1}{2\sqrt{x^3}}$$

**EXAMPLE **10

**EXAMPLE**

Determine the derivative of $latex f(x)=\frac{2}{3x^2}+ \frac{1}{\sqrt[3]{x^2}}- \frac{5}{x}$.

##### Solution

**Step 1:** We start by using the laws of exponents to write like this:

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

**Step 2:** We use the power rule to derive the terms:

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

$$f'(x)=-\frac{4}{3}x^{-3}+ \frac{2}{3}x^{-\frac{1}{3}}+ 5x^{-2}$$

**Step 3:** Finally, we simplify as follows:

$$f'(x)=-\frac{4}{3x^3}+ \frac{2}{3x^{\frac{1}{3}}}+ \frac{5}{x^2}$$

$$f'(x)=-\frac{4}{3x^3}+ \frac{2}{3\sqrt[3]{x}}+ \frac{5}{x^2}$$

## 5 Practice problems of sum and difference rule of derivatives

Apply the sum and difference rule of derivatives to find the derivatives of the following functions.

## See also

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

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