Derivative Formulas

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Derivative Calculator

The derivative is considered to be an important and powerful tool having many applications in the real life situations. It is used for example, to find inflection points, find local and global extrema, describe motion of objects, and solve optimization problems. In solving problems that require you to find the derivative of a function, it may be complex for you and to simplify the process, derivative calculators are used to help you find the answers to your problem or question. In this guide, we will look at what derivatives are and how to calculate the derivative of a function. We will also talk about the applications of derivatives and how the derivative calculator works.

The Concept of a Derivative 

When we talk of a derivative, it simply involves calculating the rate of change within a given function. Take for example, if you have a function describing the way a vehicle is moving from say point A to B, determining the derivative can help you find out the acceleration of the car from point A to B. The derivative will show you how slow or fast the speed of the vehicle changes.  

Derivative Function / Equation

Okay, let’s see how we can simplify a function.

It’s important to understand that functions, which aren’t simplified are still going to give the same derivative, however, sometimes, it can be quite difficult to calculate that.

If we have the equation  (5x + 6x)/2 + 16x +4, we can simplify it.

(11x)/2 +16x + 4

5.5x + 16x + 4

So, your final result will be;

21.5x + 4

When we have a function, we can identify its form in the following ways;

  • You can have a number, for example the form 4
  • You can also have a number that is multiplied by a variable without an exponent, for example the form 16x
  • You may also have a number that’s multiplied by a variable having an exponent, for example the form 16x2
  • There is also the form of addition, for example the form 16x + 4
  • You may have multiplication of variables, for example the form x times x or y times y.
  • There is also the division of variables for example the form, x/x

So, when the form is a just a number, it means that the derivative of that function is zero. For example (5)’ = 0 or ( ‘ = 0.

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The reason you have the derivative as zero is simply because you are not having any change in the function.

When you have a number and its multiplied by a variable that has no exponent, in this case, the derivative of this form of function is usually the number. For example;

(16x)’ = 16

(x)’ = 1

A point to note here is that if x has no an exponent, it means that the function is growing at an unchanged, steady, or constant rate.

When you have a number that is multiplied by a variable that has no an exponent, in this case, you will multiply that number by the exponent value and subtract one from that exponent you are given. For example;

5x4 = (5 x 4)(x(4-1)) = 20x3

Let’s use the slope formula to find the derivative of a given function. In this case, let’s have the function y = f(x)

Slope = Change in y times Change in x = ΔyΔx

When you look at the above diagram, you will find that

x changes to from x to x+Δx and y changes from f(x) to f(x+Δx).

What we do now is to apply the slope formula and simplify the equation.

ΔyΔx = f(x+Δx) − f(x)Δx

Since we know that f(x) = x2, now we can calculate f(x+Δx).

So, we will have f(x+Δx) = (x+Δx)2

When we expand this (x + Δx)2 we will get this f(x+Δx) = x2 + 2x Δx + (Δx)2

Since the slope formula is f(x+Δx) − f(x) Δx, what we do is put in the forms f(x+Δx) and f(x).

You will have x2 + 2x Δx + (Δx)2 − x2 Δx

If we continue simplifying, we will get; 

2x Δx + (Δx)2 Δx 

= 2x + Δx

= 2x

Therefore, the derivative of x2 will be 2x.

You can see, finding the derivative of a function can be quite a difficult thing. This is where derivative calculators come in handy.

What’s a Derivative Calculator?

A derivative calculator is simply a computed form of finding the derivative of a function. The calculator does this by calculating the derivative of the function in relation to the variable and derivative given.

In understanding how a derivative calculator works, you need to have some technical background.  When you have a mathematical function, the parser will analyze it and transform it into a tree, which is a form that the computer can understand. When transforming the function, the calculator respects the order of operations. For example, when you write mathematical expressions, you may find that the multiplication sign is sometimes left out. Take for instance, when you want to write 5 times x, you may just write 5x. In this case, the derivative calculation should detect those scenarios and ensure it inserts the multiplication sign. When you put the mathematical function, the calculator sends it to a server where it is analyzed and transformed into an form that a computer algebra system is able to understand.

It is the computer algebra system that does the computing of the derivative of your mathematical function. And just like many other computer algebra system, it will apply various rules to simplify the function before it calculates the derivative in accordance with the known differentiation rules. When the calculations are completed, they are then presented to you through your browser.

Application of Derivatives

You will find that if you want to find the derivative of a function, what you need to do is differentiate the function . When you have a curve of a function, usually, the derivative is the curve’s tangent. You can find the application of derivatives in different real life problems. So, you may see derivatives being frequently used in finding or calculating acceleration and velocity in applied mathematics. Derivatives are also be used in medical sciences, computer science, applied science, and in economics.

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