Derivative of e^9x

We now understand the derivative of e^9x. It is commonly represented as d/dx (e^9x) or (e^9x)', and its value is 9e^9x. The function e^9x has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Exponential Function: (e^9x) is an exponential function where the base is the mathematical constant e and the exponent is 9x.

Chain Rule: Rule for differentiating e^9x since the exponent involves a linear function of x.

Constant Multiple: The derivative involves multiplying by the constant 9 due to the linear function 9x.

The derivative of e^9x can be denoted as d/dx (e^9x) or (e^9x)'.

The formula we use to differentiate e^9x is: d/dx (e^9x) = 9e^9x (or) (e^9x)' = 9e^9x

The formula applies to all x, as exponential functions are differentiable everywhere.

We can derive the derivative of e^9x using proofs. To show this, we will use the chain rule along with the rules of differentiation. There are several methods we use to prove this, such as:

1. Using the Chain Rule
2. Using the properties of exponential functions

We will now demonstrate that the differentiation of e^9x results in 9e^9x using the above-mentioned methods:

Using the Chain Rule

To prove the differentiation of e^9x using the chain rule, We use the formula: Let y = e^9x To differentiate, we consider y = e^u where u = 9x. By the chain rule: dy/dx = dy/du * du/dx We know, dy/du = e^u = e^9x and du/dx = 9

Therefore, dy/dx = e^9x * 9 = 9e^9x

Hence, proved.

Using the properties of exponential functions

The derivative of e^x with respect to x is e^x, which is a fundamental property of exponential functions.

For e^9x, we apply the chain rule as follows: Let y = e^9x

Using the chain rule: dy/dx = 9e^9x

Here, we multiply by the derivative of 9x, which is 9, leading to the result 9e^9x.

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a little tricky.

To understand them better, think of a car where the speed changes (first derivative) and the rate at which the speed changes (second derivative) also changes. Higher-order derivatives make it easier to understand functions like e^9x.

For the first derivative of a function, we write f′(x), which indicates how the function changes or its slope at a certain point. The second derivative is derived from the first derivative, which is denoted using f′′(x). Similarly, the third derivative, f′′′(x), is the result of the second derivative, and this pattern continues.

For the nth Derivative of e^9x, we generally use fⁿ(x) for the nth derivative of a function f(x), which tells us the change in the rate of change (continuing for higher-order derivatives).

When x approaches infinity, the function e^9x grows exponentially. When x is zero, the derivative of e^9x = 9e^0, which is 9.

• Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.

• Exponential Function: A function in the form of e^x, where e is a mathematical constant.

• Chain Rule: A rule for finding the derivative of the composition of two or more functions.

• Constant Multiple Rule: A rule stating that the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.

• Quotient Rule: A rule used to find the derivative of a quotient of two functions.