Suppose you forgot the Quotient Rule for calculating (d)/(d x)((f(x))/(g(x))) . Use the Chain Ru

step 1 Okay, so F of X over G of X can be rewritten as F of X into G of X to the power minus one. And i

Suppose you forgot the Quotient Rule for calculating (d)/(d...

Suppose you forgot the Quotient Rule for calculating $\frac{d}{d x}\left(\frac{f(x)}{g(x)}\right) .$ Use the Chain Rule and Product Rule with the identity $\frac{f(x)}{g(x)}=f(x)(g(x))^{-1}$ to derive the Quotient Rule.

Key Concepts

Quotient Rule Derivation

The quotient rule provides a systematic method for differentiating a ratio of two functions. It can be derived by rewriting a quotient as the product of the numerator and the inverse of the denominator. By applying both the product rule and the chain rule (with the power rule for the inverse function), one obtains the general formula for the derivative of a quotient, which expresses the result in terms of the derivatives of the numerator and the denominator.

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that if a function is composed of an outer function and an inner function, the derivative of the whole function is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function. This rule is essential when dealing with functions where one function is nested inside another.

Product Rule

The product rule is a key method for differentiating the product of two functions. According to this rule, if two functions are multiplied together, the derivative of the product is the first function times the derivative of the second function plus the second function times the derivative of the first function. This rule is pivotal when one expresses a quotient as a product with an inverse.

Power Rule for Negative Exponents

The power rule allows for the differentiation of functions of the form x^n, where n can be any real number, including negative values. When applied to functions raised to a negative exponent, such as an inverse function, it requires multiplying by the exponent and then reducing the exponent by one. This differentiation technique is used in tandem with the chain rule when the base of the exponent is itself a function.

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