The second derivative of a function can be found by differentiating its first derivative. This means that the second derivative is calculated by differentiating the function twice. The second derivative has important applications, among them, it allows us to find inflection points of the function.

Here, we will learn how to find the second derivative of a function. We will learn about the process that we can use, and we will apply it to solve some practice problems.

## Process to find the second derivative of a function

The derivative of $latex \frac{dy}{dx}$, that is, $latex \frac{d}{dx}(\frac{dy}{dx})$, is denoted by $latex \frac{d^ 2y}{dx^2}$ and it’s called the second derivative of *y* with respect to *x*.

Similarly, the derivative of $latex f'(x)$ is denoted $latex f^{\prime \prime}(x)$ and is called the second derivative of $latex f(x)$ with respect to *x*.

To find the second derivative of a function, we have to differentiate the function twice. For example, suppose we want to find the second derivative of the following function:

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

For this, we follow the following steps:

**Step 1:** When we have radicals or rational expressions, we write them in exponential form by using the laws of exponents. In this case, we have:

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

**Step 2:** Find the first derivative of the function using the power rule or other applicable rules. In this case, we use the power rule on both terms of the function:

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

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

**Step 3:** Now, we differentiate $latex f'(x)$ using any applicable rules. In this case, we use the power rule again:

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

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

**Step 4:** Simplify the resulting expression. In this case, we use the laws of exponents to write as follows:

$$f^{\prime \prime}(x)=\frac{2}{x^3}$$

## Examples of the second derivative of a function

The process seen above is used to find the second derivative in the following examples. Each example has its respective detailed solution.

**EXAMPLE 1**

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

##### Solution

**Step 1:** In this case, we only have a numerical exponent, so we don’t have to simplify.

**Step 2:** We find the first derivative using the power rule:

$latex f(x)=x^3$

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

**Step 3:** We use the power rule again to derive $latex f'(x)$:

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

$latex f^{\prime \prime}(x)=6x$

**Step 4:** The expression is now simplified.

**EXAMPLE **2

**EXAMPLE**Find the second derivative of the function $latex f(x)=5x^4-3x^2+5x$.

##### Solution

**Step 1:** We don’t have to simplify, since we only have numerical exponents.

**Step 2:** We find the first derivative of the function as follows:

$latex f(x)=5x^4-3x^2+5x$

$latex f'(x)=20x^3-6x+5$

**Step 3:** We find the second derivative by differentiating $latex f'(x)$:

$latex f'(x)=20x^3-6x+5$

$latex f^{\prime \prime}(x)=60x^2-6$

**Step 4:** The expression is now simplified.

**EXAMPLE **3

**EXAMPLE**Determine the second derivative of the function $latex f(x)=10x^7+\frac{2}{x}$.

##### Solution

**Step 1:** In this case, we can use the laws of exponents to write as follows:

$$f(x)=10x^7+\frac{2}{x}$$

$latex f(x)=10x^7+2x^{-1}$

**Step 2:** Now we can use the power rule to differentiate the function:

$latex f(x)=10x^7+2x^{-1}$

$latex f'(x)=70x^6-2x^{-2}$

**Step 3:** We use the power rule again to differentiate $latex f'(x)$:

$latex f'(x)=70x^6-2x^{-2}$

$latex f^{\prime \prime}(x)=420x^5+4x^{-3}$

**Step 4:** We can simplify by using the laws of exponents and writing as follows:

$latex f^{\prime \prime}(x)=420x^5+4x^{-3}$

$$f^{\prime \prime}(x)=420x^5+\frac{4}{x^3}$$

**EXAMPLE **4

**EXAMPLE**Differentiate the function until you find its second derivative: $latex f(x) = -2x^{-5}+\sqrt{x}$.

##### Solution

**Step 1:** We simplify the function by writing the square root as a numerical exponent:

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

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

**Step 2:** We find the first derivative as follows:

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

$$f'(x)=10x^{-6}+\frac{1}{2}x^{-\frac{1}{2}}$$

**Step 3:** We differentiate $latex f'(x)$ to find the second derivative:

$$f'(x)=10x^{-6}+\frac{1}{2}x^{-\frac{1}{2}}$$

$$f^{\prime \prime}(x)=-60x^{-7}-\frac{1}{4}x^{-\frac{3}{2}}$$

**Step 4:** We use the laws of exponents to simplify as follows:

$$f^{\prime \prime}(x)=-60x^{-7}-\frac{1}{4}x^{-\frac{3}{2}}$$

$$f^{\prime \prime}(x)=-\frac{60}{x^7}-\frac{1}{4\sqrt{x^3}}$$

**EXAMPLE **5

**EXAMPLE**Find the second derivative of $latex f(x)=\sin(x)-\cos(x)$.

##### Solution

**Step 1:** We have nothing to simplify, as we only have simple trigonometric functions.

**Step 2:** We can use the rules of derivatives of trigonometric functions to derive:

$latex f(x)=\sin(x)-\cos(x)$

$latex f'(x)=\cos(x)+\sin(x)$

**Step 3:** We use the rules of derivatives of trigonometric functions again to differentiate $latex f'(x)$:

$latex f'(x)=\cos(x)+\sin(x)$

$latex f^{\prime \prime}(x)=-\sin(x)+\cos(x)$

**Step 4:** The expression is now simplified.

**EXAMPLE **6

**EXAMPLE**Find the second derivative of $latex f(x)=\frac{1}{\sqrt{x}}+3x^{-3}+5$.

##### Solution

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

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

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

**Step 2:** Using the power rule, we derive as follows:

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

$$f'(x)=-\frac{1}{2}x^{-\frac{3}{2}}-9x^{-4}$$

**Step 3:** We use the power rule again to differentiate $latex f'(x)$:

$$f'(x)=-\frac{1}{2}x^{-\frac{3}{2}}-9x^{-4}$$

$$ f^{\prime \prime}(x)=\frac{3}{4}x^{-\frac{5}{2}}+36x^{-5}$$

**Step 4:** We can simplify by using the laws of exponents and writing as follows:

$$ f^{\prime \prime}(x)=\frac{3}{4}x^{-\frac{5}{2}}+36x^{-5}$$

$$ f^{\prime \prime}(x)=\frac{3}{4x^{\frac{5}{2}}}+\frac{36}{x^5}$$

$$ f^{\prime \prime}(x)=\frac{3}{4\sqrt{x^5}}+\frac{36}{x^5}$$

## Second derivative of a function – Practice problems

Use the process above to find the second derivative of functions and solve the following practice problems.

## See also

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

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