- Calculus
- Derivatives

Explore Derivatives

- Derivatives of functions
- Derivatives of expressions
- First derivatives

- Second derivatives
- Higher-order derivatives
- Derivative of an integral

If you’ve got some questions about derivatives — like what they are, why they’re used, and how you can find one — you’ve come to the right place! Derivatives are a fundamental concept on your calculus journey, so don’t hesitate to really spend some time on this.

We don’t want to be overzealous, but we think you’ll *derive* a lot of value from the details below (sorry, we had to).

Anyway, enough with the jokes! Time to focus on our strengths.

Let’s learn some math, shall we?

## What is a derivative?

A derivative is one of the most important tools in differential calculus. Derivatives give us a flexible way to measure precise rates of change, which is really cool!

You can use derivatives in many different scenarios, including:

- Derivatives of functions
- Derivatives of expressions
- Derivatives of integrals
- First derivatives & second derivatives
- Higher-order derivatives

### The meaning of derivatives

To put it simply, derivatives show us the instantaneous rate of change at a particular point on the graph of a function. That means we’re able to capture a pretty robust piece of information with relative ease (depending on the level of calculus you’re performing!).

Plus, being able to find derivatives gives us the ability to more accurately model things like velocity, force, acceleration, and more — so they’re not just used to challenge you at homework time. Derivatives are actually crucial to the inner workings of so many things around us.

For example: let’s say you’re at a baseball game, and you want to know the rate of change of the ball’s speed as it leaves the pitcher’s hand. In that case, you’d want to take a derivative!

### Derivative and slope

It’s hard to talk about derivatives without relating them to slope.

Why? Because finding a derivative is actually equivalent to finding the slope of the tangent line at a particular point on a function.

Fun fact: How we calculate a derivative is based on how we calculate slope! It’s rise over run, but with a few interesting twists.

## The definition of derivative

So, knowing the context of derivatives and what they tell us, how are we defining “derivative”? This is our definition:

**Derivative**: (*n*) the rate of change of a quantity with respect to a change in a variable; the result of differentiation

Simple enough, right?

## Derivatives in math vs. derivatives in finance

To be clear, we’re here to teach you about derivatives in math, but you may also come across information regarding derivatives in finance or investing.

If that’s what you’re looking for, you’ll want to know that a “derivative” in finance is a contract whose value is derived from the performance of an asset, interest rate, or other “underlying.”

But we’re here to learn math, so let’s get into some equations!

## What will equations with derivatives look like?

We can talk about the “why” of derivatives until we’re blue in the face, but now it’s time to focus on the “how” and take a look at what derivatives will look like on the page.

Depending on whether you’re taking the derivative of a function, integral, or expression, your exact course of action will be different. But your starting problems will probably look something like these examples:

$$f(x)=6x^5+x$$

$$\frac{d}{dx}\int_{\frac{\pi}{4}}^{{x}} \cos(t)dt$$

$$\frac{d}{dx}\int_{-x}^{x} \ln{(2+\sin{t})}dt$$

And you’ll find yourself with solutions that look like this:

$$f^{'}(x)=4x+3$$

$$\frac{dy}{dx}=\frac{98x}{y\times(x^2+49)^2}$$

*Note: Those are just examples, NOT the solutions to those problems!*

While you’re navigating these equations, don’t forget about the differentiation rules that can help you manipulate calculations more strategically!

Constant multiple property of derivatives | $$\frac{d}{dx}\left(c\times f(x)\right)=c\times\frac{d}{dx}\left(f(x) \right)$$ |

Sum rule for derivatives | $$\frac{d}{dx}\left(f(x) + g(x)\right)=\frac{d}{dx}\left( f(x) \right)+\frac{d}{dx}\left( g(x) \right)$$ |

Difference rule for derivatives | $$\frac{d}{dx}\left(f(x) - g(x)\right)=\frac{d}{dx}\left( f(x) \right)-\frac{d}{dx}\left( g(x) \right)$$ |

Product rule for derivatives | $$\frac{d}{dx}\left(f(x)\times g(x)\right)=\frac{d}{dx}\left( f(x) \right)\times g(x)+f(x)\times\frac{d}{dx}\left( g(x) \right) $$ |

Quotient rule for derivatives | $$\frac{d}{dx}\left(\frac{f(x)}{g(x)} \right)=\frac{\frac{d}{dx}\left(f(x) \right)\times g(x)-f(x)\times\frac{d}{dx}\left( g(x) \right)}{(g(x))^{2}}$$ |

The Chain rule | $$(f\circ g)^{\prime}(x)=f^{\prime}(g(x))\times g^{\prime}(x) \text{ or } \frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx} \text{ when } y=f(u), u=g(x)$$ |

Derivative of the inverse function | $$\left(f^{-1}\right)^{\prime}(x)=\frac{1}{f^{\prime}\left(f^{-1}\left(x\right)\right)}$$ |

As you can tell, there’s quite a variety of derivative equations, so let’s get a little more specific!

### Derivative of a function

Finding derivatives of functions is usually the starting point of your adventure with differentiation, so be sure to master those procedures before moving on to limits and more challenging differentiation tasks.

As with anything in life, it’s helpful to keep your goals in mind while you work! So, what is our goal with derivatives of functions?

When we take the derivative of a function, it’s because we want to find:

- The function’s rate of change
- The slope of the tangent line at a particular point on the function

We’ll go into how to do this in a moment; but first, let’s decode something else you may hear when learning derivatives of functions.

### Derivative with respect to x

Maybe you’ve heard your math teacher talk about a derivative “with respect to” a variable (at the beginning, it’s commonly $$x$$), and you’re not quite sure what that really means.

If we’re working with a function — $$f(x)$$ — and we want to find “the derivative with respect to $$x$$,” that just means we’re looking to find the rate at which $$f$$ changes as $$x$$ changes. Essentially, it means we’re really focusing in on the variable $$x$$ and how it affects our function.

### The process of finding the derivative of a function

If you remember your key vocabulary terms, you’ll remember that the process of finding the derivative is called “differentiation.” (We can help if you need a refresher on calculus vocab!)

Fun fact: the derivative of a function is also a function, and its values are the corresponding derivatives at a point. Essentially, if we determine the derivative of a function $$f$$ at any value $$x$$, then our derivative $$f'(x)$$ represents a function that outputs the value of $$f'(x)$$ for any input $$x$$ (at least, for which this derivative exists).

The actual steps of differentiation will vary based on the individual problem and the complexity — we can explain in detail in the app! — but here’s an overview:

- Take the derivative on both sides of the equation
- Use the differentiation rules (also called “derivative formulas”)
- Find the derivative
- Simplify the expression, if needed

Need more? Hop on over to dive deeper into derivatives of functions.

Don’t forget: You can also find the derivative of expressions and integrals, which will look a little different.

## What is f’(x)?

$$f’(x)$$ denotes the derivative. So, while you start taking the derivative of a function $$f(x)$$, your solution — the derivative — will be labeled as $$f’(x)$$. You may also see $$f’(x)$$ written as $$\frac{dy}{dx}$$ in certain scenarios.

Just be careful when reading, because it’s easy to miss that little apostrophe. When we see $$f’(x)$$, we’re seeing a derivative, but $$f(x)$$ tells us it’s a function!

## Derivative vs. limit: What’s the difference?

Derivatives and limits aren’t interchangeable terms, but there is some overlap.

The difference depends on whether you’re talking about derivatives and limits of a certain value of a function, or of the function as a whole. Let us explain:

- A derivative of a function at a point is a special type of limit at that point — in other words, every derivative is a limit!
- Numerically, the derivative’s value at a certain point of a function tells us the function’s instantaneous rate of change at that point.
- Geometrically, the derivative’s value at a certain point of a function represents the slope of a line running tangent to the function’s graph at that point.

- The limit of a function at a point represents the behavior of the function around that point. Basically, if the behavior is predictable, then the limit exists.

Does that help you *differentiate*? (Pun very much intended!)

## Examples of derivatives in math

You know we’re big advocates of practice, so let’s look at some example problems! Using what you’ve learned, you can work through these examples on your own.

- $$f(x)=e^x+10x$$
- $$f(x)=\frac{x-3}{\sqrt{8x^2-2}}$$
- $$\frac{d}{dx}\left(\sqrt{x^2-3} \right)$$
- $$\frac{d}{dx}\left(\ln{(x)}+7^x\right)$$
- $$\frac{d}{dx}\int_{x}^{\frac{\pi}4} (t^2-\ln(t))^2dt$$

**Here’s how we solve the first practice problem in the app:**

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Stuck on the other derivative practice problems? Scan the problem with your Photomath app so we can walk you through each step!

**FAQ***What are the two definitions of a derivative?*

A derivative is described as either the rate of change of a function, or the slope of the tangent line at a particular point on a function.

*What is a derivative in simple terms?*

A derivative tells us the rate of change with respect to a certain variable.

*How are derivatives used in real life?*

Derivatives can be used to predict changes like temperature variation in climate change, earthquake magnitude ranges, population census predictions, shifts in momentum, and more.

I'm an enthusiast with a deep understanding of calculus, particularly derivatives. I've explored derivatives extensively and can provide valuable insights into their concepts and applications. Let's delve into the key points mentioned in the article.

**Calculus and Derivatives: A Deep Dive**

**1. What is a Derivative?**

- A derivative is a crucial tool in differential calculus, offering a flexible way to measure precise rates of change.
- It finds applications in various scenarios, including derivatives of functions, expressions, and integrals.

**2. Meaning of Derivatives:**

- Derivatives show the instantaneous rate of change at a specific point on a function's graph.
- They provide essential information for modeling velocity, force, acceleration, etc., making them integral to various real-world phenomena.

**3. Derivative and Slope:**

- Calculating a derivative is equivalent to finding the slope of the tangent line at a particular point on a function.
- The calculation is based on the concept of rise over run, but with interesting twists.

**4. Definition of Derivative:**

- Derivative is defined as the rate of change of a quantity with respect to a change in a variable; the result of differentiation.

**5. Equations with Derivatives:**

- Examples of equations involving derivatives are provided, covering functions, integrals, and expressions.
- Solutions to these examples are presented, emphasizing the importance of differentiation rules.

**6. Differentiation Rules:**

- Constant multiple property, sum rule, difference rule, product rule, quotient rule, and the chain rule are explained.
- These rules facilitate strategic manipulation of derivative calculations.

**7. Derivative of a Function:**

- Finding derivatives of functions is highlighted as the starting point in the journey of differentiation.
- The goals include determining the function's rate of change and the slope of the tangent line at a specific point.

**8. Derivative with Respect to x:**

- Understanding derivatives "with respect to" a variable, commonly denoted as $$x$$, involves focusing on how $$f$$ changes as $$x$$ changes.

**9. Derivative vs. Limit:**

- Derivatives and limits are related but not interchangeable. Derivatives are a special type of limit, representing instantaneous rate of change.

**10. Examples of Derivatives in Math:**

- Practice problems are provided, encouraging readers to apply their knowledge to functions, expressions, and integrals.

**11. Practical Applications:**

- Derivatives are not just theoretical; they have real-life applications, such as predicting changes in climate, earthquake magnitudes, population census, and more.

In summary, derivatives are a fundamental concept in calculus, offering a powerful tool to understand and model changes in various fields. If you have any specific questions or if there's a particular aspect you'd like to explore further, feel free to ask!