Suppose f and g are functions that are differentiable at x=1...

Suppose $f$ and $g$ are functions that are differentiable at $x=1$ and that $f(1)=2, f^{\prime}(1)=-1, g(1)=-2,$ and $g^{\prime}(1)=3 .$ Find the value of $h^{\prime}(1)$.
$$h(x)=\frac{f(x) g(x)}{f(x)-g(x)}$$

Key Concepts

Quotient Rule

The quotient rule is applied to differentiate a function that is expressed as the ratio of two differentiable functions. According to the rule, the derivative of a quotient is determined by taking the derivative of the numerator multiplied by the denominator, subtracting the numerator multiplied by the derivative of the denominator, and dividing the result by the square of the denominator. This rule is critical for correctly finding the derivative of rational functions.

Differentiability

Differentiability is a property of a function indicating that its derivative exists at a given point or over an interval. This concept is fundamental to calculus because it allows us to understand how functions change and to apply standard rules of differentiation.

Product Rule

The product rule is used to find the derivative of a function that is the product of two or more differentiable functions. It states that the derivative is the sum of each function's derivative multiplied by the other function(s), which is essential when handling expressions involving function multiplication.

Evaluation of Derivatives at a Specific Point

Evaluating a derivative at a specific point involves substituting that point into the derivative expression. This technique is used to calculate the instantaneous rate of change of a function at that exact point, providing insight into the behavior of the function at that location.

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