Derivative of Area of Triangle

We now understand the derivative of the area of a triangle. It is commonly represented as d/dx (Area) and involves differentiating the standard area formula with respect to one of the triangle's variables. The area of a triangle is given by (1/2) * base * height. Differentiating this function with respect to its variables allows us to analyze how changes in the base or height affect the area.

The key concepts are mentioned below: 

Triangle Area Formula: Area = (1/2) * base * height. 

Product Rule: Used when differentiating the product of base and height.

Chain Rule: Used in cases where base or height is a function of another variable.

The derivative of the area of a triangle can be denoted as d/dx (Area).

The formula we use to differentiate the area is: d/dx (Area) = (1/2) * (b' * h + b * h')

Where b and h are the base and height, respectively, and b' and h' are their derivatives with respect to x.

This formula applies to all x where both base and height are differentiable.

We can derive the derivative of the area of a triangle using proofs. To show this, we will use the rules of differentiation.

There are several methods we use to prove this, such as: 

• By First Principle 
   
• Using Product Rule

We will now demonstrate that the differentiation of the area results in the formula mentioned above using these methods:

By First Principle

The derivative of the area can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative using the first principle, consider f(x) = (1/2) * b * h. Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h Given that f(x) = (1/2) * b * h, we write f(x + h) = (1/2) * (b(x + h)) * (h(x + h)). Substituting these into the equation, f'(x) = limₕ→₀ [(1/2) * (b(x + h)) * (h(x + h)) - (1/2) * b * h] / h Expanding and simplifying will lead to the derivative formula.

Using Product Rule

To prove the differentiation of the area of a triangle using the product rule, We use the formula: Area = (1/2) * b * h By the product rule: d/dx [b * h] = b' * h + b * h' Therefore, d/dx (Area) = (1/2) * (b' * h + b * h')

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be complex to understand. Consider a scenario where not only the base or height changes but also the rate at which they change over time. Higher-order derivatives make it easier to understand such dynamic changes.

For the first derivative of the area, we write f′(x), indicating how the area changes with respect to x. The second derivative is derived from the first derivative, denoted using f′′(x), and this pattern continues.

For the nth Derivative of the area, we generally use fⁿ(x) to indicate the change in the rate of change.

When either the base or height approaches zero, the derivative will reflect a rapid change as the area approaches zero.

When the base or height is constant, the derivative with respect to that variable will be zero, indicating no change in area with respect to that variable.

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

•  Triangle Area Formula: A formula that calculates the area of a triangle using base and height. 

• Product Rule: A differentiation rule used when differentiating a product of two functions.

• First Derivative: The initial result of differentiating a function, showing the rate of change of a specific function. 

• Chain Rule: A differentiation rule used for functions composed of other functions, vital when variables depend on other variables.