Introduction to the derivative of x^3

Derivatives have a wide range of applications in almost every field of engineering and science. The differentiation of x^3 can be calculated by following the rules of differentiation. Or, we can directly find the derivative formula of x^3 by applying the first principle of differentiation. In this article, you will learn what the x^3 derivative formula is and how to calculate the x cube derivative by using different approaches.

What is the derivative of x^3?

The derivative of x^3 is equal to 3x^2. It is denoted by d/dx(x^3) which is a fundamental concept in calculus. It measures the rate of change of the algebraic function x^3. 

Knowing the formula for derivatives and understanding how to use it can be used in solving problems related to velocity, acceleration, and optimization. 

Derivative of x^3 formula

The formula for derivative of f(x)=x^3 is equal to the 4x^3, that is;

$f'(x) = \frac{d}{dx} (x^3) = 3x^2$

It is calculated by using the power rule of differentiation, which is defined as:

$\frac{d}{dx}(x^n)=nx^{n-1}$

How do you differentiate x^3?

There are multiple ways to prove the differentiation of x3. These are;

1. First Principle
2. Product Rule
3. Quotient Rule

Each method provides a different way to compute the x^3 differentiation. By using these methods, we can mathematically prove the formula for finding the derivative of x 3.

Derivative of x power 3 by first principle

According to the first principle of derivative, the differentiation of x^3 is equal to 3x^2. The derivative of a function by first principle refers to finding a general expression for the slope of a curve by using algebra. It is also known as the delta method. The derivative is a measure of the instantaneous rate of change, which is equal to,

$f’(x) = \lim_{h\to 0}\frac{f(x + h) - f(x)}{h}$

This formula allows us to determine the rate of change of a function at a specific point by using the limit definition of the derivative.

Proof of x^3 derivative formula by first principle

To prove the derivative of x3 by using the first principle, replace f(x) with x^3 or you can replace it with x^4 to find the derivative of x^4. 

$f(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

$f'(x) = \lim_{h \to 0} \frac{(x + h)^3 - x^3}{h}$

Moreover,

$f'(x) = \lim_{h \to 0} \frac{x^2 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4 - x^4}{h}$

$f'(x) = \lim_{h \to 0} \frac{4x^3h + 6x^2h^2 + 4xh^3 + h^4}{h}$

When h approaches to zero,

$f’(x)=3x^2$

Hence the differentiation of x^3 is equal to 3x^2. Also calculate the derivative of x^3 at a specific point by using our derivative at a point calculator.

Derivative of x cube by product rule

The formula for derivative x^3 can be calculated by using the product rule because an algebraic function can be written as the combination of two functions. The product rule derivative is defined as;

$[uv]’ = u’.v + u.v’$

Proof of x cube derivative by product rule

To differentiate of x^3 by using the product rule, we start by assuming that,

$f(x) = x^2. x$

By using product rule of differentiation,

$f’(x) = (2x). x + (x^2)(1)$

We get,

$f’(x) = 2x^2 + x^2$

Hence,

$f’(x) = 3x^2$

Also use our product rule calculator online, as it provides you with a step-by-step solution of the differentiation of a function.

Derivative of x3 using quotient rule

Another method for finding the derivative of x 3 is using the quotient rule, which is a formula for finding the derivative of a quotient of two functions. Since the secant function is the reciprocal of cosine, the derivative of cosecant can also be calculated using the quotient rule. The derivative quotient rule is defined as:

$\frac{d}{dx}\left(\frac{f}{g}\right) = \frac{f’(x). g(x) - g’(x).f(x)}{(g(x))^2}$

Proof of differentiating x3 by quotient rule

To prove the x^3 derivative, we can start by writing it,

$f(x) = \frac{x^4}{1}= \frac{u}{v}$

Supposing that u = x^3 and v = 1. Now by quotient rule,

$f’(x) = \frac{vu’ - uv’}{v^2}$

$f’(x) = \frac{1.(3x^2) - x^3.(0)’}{(1)^2}$

$f’(x)= \frac{2x^3}{1}$

$f’(x)=3x^2$

Hence, we have derived the derivative of x^3 using the quotient rule of differentiation. Use our quotient rule calculator with steps to find the derivative of any quotient function.

How to find the derivatives of x^3 with a calculator?

The easiest way of differentiating x^3 is by using an online calculator. You can use our differential calculator for this. Here, we provide you a step-by-step way to calculate derivatives by using this tool.

1. Write the function as x^3 in the enter function box. In this step, you need to provide input value as a function that you want to differentiate.
2. Now, select the variable by which you want to differentiate x^3. Here you have to choose x.
3. Select how many times you want to calculate the derivatives of x^3. In this step, you can choose 2 to calculate the second derivative, 3 for the third derivative and so on.
4. Click on the calculate button. After this step, you will get the derivative of x cube within a few seconds.

After completing these steps, you will receive the differential of x3 within seconds. Using online tools like third derivative calculator, can make it much easier and faster to calculate derivatives, especially for complex functions.

Conclusion:

In conclusion, the derivative of x3 is 3x^2. The derivative measures the rate of change of a function with respect to its independent variable, and in the case of x^3, the derivative can be calculated using the power rule, product rule and quotient rule of differentiation.