Introduction to the Derivative of e^8x

Derivatives have a wide range of applications in almost every field of engineering and science. The derivative of e can be calculated by following the rules of differentiation. Or, we can directly find the derivative of e8 by applying the first principle of differentiation. In this article, you will learn what the derivative of e8 is and how to calculate the derivative of e8x by using different approaches.

What is the derivative of e^(8x)?

The derivative of e^8x is equal to 8e raised to the power of 8x and is denoted by d/dx (e8x). This represents the rate of change of the function at any given point.

Calculating the derivative of e to the 8 with respect to x is a fundamental concept in calculus. Remarkably, the derivative of e raised to the power of x is itself, and in this case, the constant coefficient is 8.

Derivative of e8x formula

The formula for the derivative of e raised to the power of 8x is equal to the exponential function e, specifically:

d/dx (e8x) = 8e8x

This means that the rate of change of the exponential function e^8x at any given point is equal to 8 times the function itself.

How do you prove the derivative of e8x?

There are different ways to derive derivatives of e8x. Three common ways are:

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

Each method provides a different way to compute the e^8 derivative. By using these methods, we can mathematically prove the formula for finding the derivative of e^(8x).

Derivative of e8x by first principle

According to the first principle of derivative, the ln e^8x derivative is equal to 8e^8x. 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 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 limit definition of derivative.

Proof of derivative of e8x by first principle

To prove the derivative of e by using first principle, we start by replacing f(x) by e.

f'(x)=limh→0f(x+h)-f(x)/h

f'(x) = lim e8(x+h) - e8x/h

Moreover,

f'(x) = lim e8x.e8h - e8x/h

Taking e8x common as;

f'(x) = lim e8x(e8h - 1)/h

More simplification,

f'(x) = 8e8x .lim (e8h - 1)/8h

When h approaches to zero,

f'(x) = 8e8x lim (e0 - 1)/8h

f'(x) = 8e8x f(0)

Therefore,

f'(x) = 8e8x

The derivative of e squared can also be calculated by using the first principle of differentiation.

Derivative of e^(8x) by product rule

Another method to find the derivative e^8x is the product rule formula which is used in calculus to calculate the derivative of the product of two functions. Specifically, the product rule is used when you need to differentiate two functions that are multiplied together. The formula for the product rule calculator is:

d/dx(uv) = u(dv/dx) + (du/dx)v 

In this formula, u and v are functions of x, and du/dx and dv/dx are their respective derivatives with respect to x.

Proof of derivative of e8x by product rule

To prove the derivative of e by using product rule, we start by assuming that,

f(x) = e4x. e4x

By using product rule of differentiation,

f(x) = (e4x). e4x + (e4x)e4x

We get,

f(x) = 4e8x + 4e8x

Hence,

f(x) = 8e8x

Derivative of e8x using quotient rule

Since the exponential function can be written in a fraction, therefore, the derivative of e^8x can also be calculated by using the quotient rule. The quotient rule calculator formula is defined as;

d/dx (f/g) = f(x). g(x) -g(x).f(x) /{g(x)}2

Proof of derivative of e8x by quotient rule

To prove the derivative of e^8x, we can start by writing it,

f(x) = e8x /1 = u/v

Supposing that u = e8x and v = 1. Now by quotient rule,

f(x) = (vu - uv)/v2

f(x) = [d/dx(e8x) - e8x .d/dx(1)] / (1)2

= [8e8x] / 1

= 8e8x

Hence, we have derived the derivative of e8x using the quotient rule of differentiation.

How to find the e^8x derivative with a calculator?

The easiest way to calculate the derivative of e is by using an online tool. You can use our derivative calculator for this. Here, we provide you a step-by-step way to calculate derivatives by using this tool.

1. Write the function as e8x in the “enter function” box. In this step, you need to provide input value as a function as you have to calculate the derivative of e8x.
2. Now, select the variable by which you want to differentiate e8x. Here you have to choose x.
3. Select how many times you want to differentiate e8x to the x. In this step, you can choose 2 for the second derivative, 3 for the third derivative and so on.
4. Click on the calculate button.

After completing these steps, you will receive the e^8x derivative within seconds. Using online tools can make it much easier and faster to calculate derivatives, especially for complex functions.

Frequently asked questions 

What are three exponent rules?

According to the first law, we can multiply two exponential functions with the same base by simply adding their exponents. The second law states that we subtract the exponents when dividing two exponential functions with the same base. The third law states that we multiply the exponents to raise a power to a new power.

What is the derivative of e^8x?

The derivative of e8x with respect to x is 8e8x. Mathematically, the derivative of e to the x is written as;

d/dx (e8x) = 8e8x