Derivative of 5e^x

We now understand the derivative of 5e^x. It is commonly represented as d/dx (5e^x) or (5e^x)', and its value is 5e^x. The function 5e^x has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below: Exponential Function: (e^x is the base of natural logarithms).

Constant Multiple Rule: A rule for differentiating a constant times a function.

Exponential Rule: The derivative of e^x is e^x.

The derivative of 5e^x can be denoted as d/dx (5e^x) or (5e^x)'.

The formula we use to differentiate 5e^x is: d/dx (5e^x) = 5e^x (or) (5e^x)' = 5e^x

The formula applies to all x in the real number domain.

We can derive the derivative of 5e^x using proofs. To show this, we will use the rules of differentiation. There are several methods we use to prove this, such as:

1. By First Principle
2. Using Constant Multiple Rule
3. Using Exponential Rule

We will now demonstrate that the differentiation of 5e^x results in 5e^x using the above-mentioned methods:

By First Principle

The derivative of 5e^x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of 5e^x using the first principle, we will consider f(x) = 5e^x. Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1)

Given that f(x) = 5e^x, we write f(x + h) = 5e^(x + h).

Substituting these into equation (1), f'(x) = limₕ→₀ [5e^(x + h) - 5e^x] / h = 5 limₕ→₀ [e^x (e^h - 1)] / h

We know that e^h - 1 is approximately h for small h, f'(x) = 5e^x limₕ→₀ h / h = 5e^x

Hence, proved.

Using Constant Multiple Rule

To prove the differentiation of 5e^x using the constant multiple rule, We use the formula: d/dx (cf(x)) = c d/dx f(x) Let c = 5 and f(x) = e^x. d/dx (5e^x) = 5 d/dx (e^x)

Since the derivative of e^x is e^x, = 5e^x

Using Exponential Rule

The exponential rule states that the derivative of e^x is e^x. For the function 5e^x, the derivative is: d/dx (5e^x) = 5 d/dx (e^x) = 5e^x

Thus, the derivative of 5e^x is 5e^x.

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 5e^x.

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 5e^x, it remains the same, as the function is an exponential function with a constant coefficient.

The derivative of 5e^x is always 5e^x, regardless of the value of x, as the exponential function is defined for all real numbers. Since exponential functions do not have asymptotes, the derivative does not have undefined points.

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

• Exponential Function: A mathematical function of the form f(x) = e^x, where e is the base of the natural logarithm.

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

• Exponential Rule: The derivative of e^x is e^x.

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