Derivatives: Examples and Explanations
Derivatives are an integral part of calculus and mathematics overall. Finding the derivative is different for different types of equations, and it requires the knowledge of various rules, formulas, and procedures to understand completely. There are several types of derivatives for you to take. The derivative of a polynomial is a simple process and it only requires you to follow the basic rules of mathematics and derivatives to understand it, but taking the derivative of a function can get far more complicated when you involve exponents, logarithms, and trigonometric functions. A lot of people believe that derivatives are a difficult concept to grasp, but they aren’t once you learn how to do it. Understanding derivatives can be easy if you just take a little bit of time to look at how each problem is done.
Learning how to differentiate the function or take the derivative will be as important a component to your academic success as learning to write the best paper (learn more in this course). As difficult as it may seem at first, with a few examples you can figure out how to take the derivative of any function that is placed before you.
Understanding How Derivatives are Found
Above is the basic definition of derivatives. It helps explain the concept behind derivatives and why they work. The problem simply says that the derivative of a function (x) is f(x+h) – f(x)/h. In order to solve the equation you would first pick a function and then switch f(x) out with it.
An example of this working would be seeing how 4x is the derivative of 2x2. First you remove the 2 from the equation that way it becomes 2 times the derivative of x2. Then you start to work out the equation and it turns into (x + h)2– x2 all divided by h.
Multiply the x + h function by itself (x + h) (x + h) in order to get x2 + xh + xh + h2 – x2 and again divide everything by h.
Combine your like terms to get the following (2xh + h2)/h
Now pull out the h from the numerator since it’s a like term so you can get h(2x + h)/h
Once you do this you’re almost done with the equation; simply divide the h and leave yourself with 2x + h. Now the limit of a derivative is 0 so replace the h with 0 and finally the answer ends up being 2x.
That is, until you multiply it by the 2 you originally pulled out of the equation before you did the problem. This turns 2x into 4x, which is the final answer to the problem.
This is an explanation of why 2x^2 = 4x.
Although this is one type of way to derivative problems, it is certainly not the simplest. There are ways to drastically cut down on the number of steps it take to complete a derivative problem.
The Udemy course Calculus I essentials helps point out some of the more straightforward ways to deal with calculus derivatives.
Doing Derivatives the Easy Way
It’s always important to learn why things work instead of just knowing how they work. The derivative equation above 2x2 could have been solved simply by moving the exponent down multiplying it by the equation and then finally subtracting one from the equation. The problem works like so:
2x2 = 2 * 2x2 – in this step the derivative is moved down to be multiplied by the equation in this way we get 4x2.
Then we subtract 1 from the original exponent, which is 2 in this case. Therefore the problem ends up being 4x.
This is the simplest way possible to do derivative equations and it becomes an important step once you start dealing with problems that force you do things, such as the chain rule, product rule, and quotient rule. As derivative equations grow more and more complicated, it’s important to find steps that teach you how to streamline the process. An example of a problem that fully utilizes this step would be something like the problem below.
2(3x + 5)2
In order to do this problem, bring the 2 down and subtract 1 from the original exponent. Now you’re halfway done with 4(3x + 5). Finish the problem by taking the derivative of the equation inside the parenthesis, this is known as the chain rule, and multiplying that by the problem you see above. So the derivative of 3x + 5 * 4(3x+ 5) would be your answer, which turns out to be 12(3x + 5).
As learning calculus gets more complex you can use courses on Udemy such as Calculus II essentials to keep you ahead of the game. You can also look at A Career in Engineering to learn about the types of jobs you can get if you succeed at mastering calculus and derivatives
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