Derivative of 6e^x

We now understand the derivative of 6e^x. It is commonly represented as d/dx (6e^x) or (6e^x)', and its value is 6e^x.

The function 6e^x has a clearly defined derivative, indicating it is differentiable over all real numbers. The key concepts are mentioned below:

Exponential Function: e^x is the base of natural logarithms and has unique properties.

Constant Multiplication Rule: Rule for differentiating functions with constant coefficients.

Exponential Rule: This rule applies to derivatives of exponential functions.

The derivative of 6e^x can be denoted as d/dx (6e^x) or (6e^x)'.

The formula we use to differentiate 6e^x is: d/dx (6e^x) = 6e^x (or) (6e^x)' = 6e^x This formula applies to all x.

We can derive the derivative of 6e^x using the standard rules of differentiation. To show this, we will use the exponential rule along with constant multiplication. There are several methods we use to prove this, such as:

• By First Principle
   
• Using the Exponential Rule
   
• Using Chain Rule

We will now demonstrate that the differentiation of 6e^x results in 6e^x using the above-mentioned methods:

By First Principle

The derivative of 6e^x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of 6e^x using the first principle, we will consider f(x) = 6e^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) = 6e^x, we write f(x + h) = 6e^{(x + h)}. Substituting these into equation (1), f'(x) = limₕ→₀ [6e^{(x + h)} - 6e^x] / h = limₕ→₀ [6ex^{(eh - 1)}] / h = 6e^x limₕ→₀ [e^h - 1] / h Using the fact that limₕ→₀ [e^h - 1] / h = 1, we have, f'(x) = 6e^x Hence, proved.

Using the Exponential Rule

To prove the differentiation of 6e^x using the exponential rule, We use the formula: d/dx (e^x) = e^x By the constant multiplication rule, d/dx (6e^x) = 6 d/dx (e^x) = 6e^x Hence, proved.

Using Chain Rule

We will now prove the derivative of 6e^x using the chain rule. The step-by-step process is demonstrated below: Here, we use the formula, d/dx (6e^x) = 6 d/dx (e^x) Given that, u = x and v = e^x Using the chain rule formula: d/dx [u.v] = u'. v + u. v' u' = d/dx x = 1 (substitute u = x) v' = d/dx (e^x) = e^x Using the chain rule, d/dx (6e^x) = 6 (1 * e^x) = 6e^x Thus, proved.

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 6e^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 6e^x, we generally use f n(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).

The derivative of 6e^x is always defined, and there are no points where it becomes undefined. At x = 0, the derivative of 6e^x = 6e⁰, which is 6.

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

• Exponential Function: A function of the form e^x, where e is the base of natural logarithms.

• Constant Multiplication Rule: A rule used to differentiate functions with constant coefficients.

• Chain Rule: A rule for differentiating compositions of functions.

• Quotient Rule: A rule for differentiating ratios of functions.