Derivative of u^x

We now understand the derivative of u^x. It is commonly represented as d/dx (u^x) or (u^x)', and its value is u^x ln(u). The function u^x has a well-defined derivative, indicating it is differentiable.

The key concepts are mentioned below:

Exponential Function: u^x represents an exponential function.

Logarithmic Function: ln(u) represents the natural logarithm of u.

Chain Rule: Used for differentiating functions involving compositions, like u^x.

The derivative of u^x can be denoted as d/dx (u^x) or (u^x)'. The formula we use to differentiate u^x is: d/dx (u^x) = u^x ln(u)

The formula applies to all x, where u is a positive constant.

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

1. Using Logarithmic Differentiation
2. Using Chain Rule
3. Using Exponential Properties

We will now demonstrate that the differentiation of u^x results in u^x ln(u) using the above-mentioned methods:

Using Logarithmic Differentiation

To find the derivative of u^x using logarithmic differentiation, we take the natural logarithm of both sides. Consider y = u^x. Taking natural log on both sides gives us ln(y) = x ln(u).

Differentiating both sides with respect to x gives us 1/y dy/dx = ln(u). Thus, dy/dx = y ln(u) = u^x ln(u).

Hence, proved.

Using Chain Rule

To prove the differentiation of u^x using the chain rule, We use the fact that u^x can be expressed as e^(x ln(u)).

Let y = e^(x ln(u)). Then, dy/dx = e^(x ln(u)) (d/dx (x ln(u))). Since d/dx (x ln(u)) = ln(u), we have: dy/dx = e^(x ln(u)) ln(u) = u^x ln(u).

Using Exponential Properties

We can also use the exponential properties to derive the derivative of u^x. Consider f(x) = u^x.

Using the property of exponential functions, we express u^x as e^(x ln(u)).

Differentiating with respect to x, we have f'(x) = e^(x ln(u)) ln(u). Thus, f'(x) = u^x ln(u).

Hence, the derivative of u^x is u^x ln(u).

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 u^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 u^x, 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 the base u is 1, the derivative is zero because 1^x is a constant function. When the base u is e, the derivative of e^x is e^x ln(e), which simplifies to e^x.

• 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 u^x, where u is a constant base.

• Natural Logarithm: The natural logarithm is the logarithm to the base e, denoted as ln.

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

• Constant Function: A function that has the same value for any input, resulting in a zero derivative.