Derivative of 9^x

We now understand the derivative of 9^x. It is commonly represented as d/dx (9^x) or (9^x)', and its value is 9^x ln(9). The function 9^x has a clearly defined derivative, indicating it is differentiable over its entire domain. The key concepts are mentioned below: Exponential Function: 9^x is an exponential function. Natural Logarithm: ln(x) is the natural logarithm function, which is the inverse of the exponential function e^x. Chain Rule: This rule is beneficial for differentiating composite functions like 9^x.

The derivative of 9^x can be denoted as d/dx (9^x) or (9^x)'. The formula we use to differentiate 9^x is: d/dx (9^x) = 9^x ln(9) The formula applies to all x.

We can derive the derivative of 9^x using various methods. To show this, we will use logarithmic differentiation and the rules of differentiation. There are several methods we use to prove this, such as: Using Logarithmic Differentiation Using the Chain Rule Using the Exponential Function Rule We will now demonstrate that the differentiation of 9^x results in 9^x ln(9) using the above-mentioned methods: Using Logarithmic Differentiation To find the derivative of 9^x, take the natural logarithm of both sides, y = 9^x: ln(y) = ln(9^x) Using the properties of logarithms, ln(y) = x ln(9) Differentiate both sides with respect to x: (1/y) dy/dx = ln(9) dy/dx = y ln(9) Substituting y = 9^x back, dy/dx = 9^x ln(9) Hence, proved. Using the Chain Rule To prove the differentiation of 9^x using the chain rule, Let y = 9^x = e^(x ln(9)) Differentiate using the chain rule: dy/dx = d/dx [e^(x ln(9))] = e^(x ln(9)) * d/dx [x ln(9)] = 9^x * ln(9) Thus, the derivative is 9^x ln(9). Using the Exponential Function Rule Using the general rule for differentiating a^x (where a is a constant), d/dx (a^x) = a^x ln(a) For 9^x, a = 9, so: d/dx (9^x) = 9^x ln(9) Thus, the derivative is 9^x ln(9).

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be more complex. 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 9^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 9^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 x = 0, the derivative of 9^x is 9^0 ln(9), which is ln(9). When x is a large negative number, the derivative approaches zero because 9^x approaches zero.

Derivative: The derivative of a function indicates the rate at which the function changes with respect to its variable. Exponential Function: A function of the form a^x, where a is a constant and x is a variable. Natural Logarithm: The logarithm to the base e, where e is an irrational constant approximately equal to 2.718281828. Chain Rule: A fundamental rule in calculus for finding the derivative of composite functions. Logarithmic Differentiation: A method of differentiating functions by taking the natural logarithm of both sides.