Carry out Maria Agnesi's proof of the Quotient Rule from her...

Carry out Maria Agnesi's proof of the Quotient Rule from her book on calculus, published in 1748 . Assume that $f, g$ and $h=f / g$ are differentiable. Compute the derivative of $h g=f$ using the Product Rule, and solve for $h^{\prime} .$

Key Concepts

Differentiability

Differentiability refers to the property of a function that allows it to have a derivative at each point in its domain. When a function is differentiable, it means that the limit of the average rate of change exists as the interval approaches zero, which provides a precise measurement of the instantaneous rate of change or slope of the tangent line at any given point.

Product Rule

The Product Rule is a fundamental theorem in differential calculus that provides a method for finding the derivative of the product of two differentiable functions. According to this rule, the derivative of a product f(x)g(x) is given by f?(x)g(x) + f(x)g?(x). This rule is crucial for breaking down more complex derivative computations into simpler parts.

Quotient Rule

The Quotient Rule is a technique in differential calculus used to differentiate a quotient of two differentiable functions. It can be derived using the Product Rule by expressing the quotient as the product of one function and the reciprocal of the other. The rule facilitates the calculation of the derivative of a function of the form h(x) = f(x)/g(x) by providing a formula that involves the derivatives and values of both f(x) and g(x).

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