Derivative of Slope

We now understand the derivative of a slope. A derivative is commonly represented as d/dx (f(x)) or (f(x))', and it represents the rate of change of a function. The slope of a function has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: Function: A mathematical relation that assigns exactly one output value for each input value. Quotient Rule: A rule for differentiating the division of two functions. Rate of Change: How a quantity changes with respect to another quantity.

The derivative of a slope involves differentiating functions to find their instantaneous rate of change. The formula for the derivative of a function f(x) is: d/dx (f(x)) = f'(x) This formula applies to all x within the domain of f(x) where the function is continuous and differentiable.

We can derive the derivative of a slope using proofs. To show this, we will use various differentiation techniques. There are several methods we use to prove this, such as: By First Principle Using Chain Rule Using Product Rule We will now demonstrate that the differentiation of a function results in its derivative using the above-mentioned methods: By First Principle The derivative can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of a function f(x) using the first principle, we write: f'(x) = limₕ→₀ [f(x + h) - f(x)] / h This method calculates the instantaneous rate of change by considering an infinitesimal change in x. Using Chain Rule To prove the differentiation of a composite function using the chain rule, We use the formula: d/dx [g(f(x))] = g'(f(x)) · f'(x) This method is applied to functions that are compositions of other functions. Using Product Rule We use the product rule to differentiate the product of two functions. The formula is: d/dx [u(x) · v(x)] = u'(x) · v(x) + u(x) · v'(x) This method is applied when a function is the product of two other functions.

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 velocity and acceleration. 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 a function f(x), we generally use fⁿ(x) for the nth derivative, which tells us the change in the rate of change.

When dealing with functions that have discontinuities or asymptotes, the derivative may be undefined at those points. For example, if a function has a vertical asymptote at x = c, the derivative is undefined there. In cases where the function is continuous and differentiable, the derivative is well-defined.

Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. Function: A mathematical relation that assigns exactly one output value for each input value. Quotient Rule: A technique used to find the derivative of a division of two functions. Chain Rule: A formula for finding the derivative of a composite function. Rate of Change: The speed at which one quantity changes with respect to another.