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 $h^{\prime}(1)$.
$$h(x)=\left(x^{2}+1\right) g(x)$$

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Key Concepts

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate at which the function’s value is changing with respect to changes in its input. It is a fundamental concept in calculus used to analyze and describe the behavior of functions, including determining local maxima and minima, and understanding instantaneous rates of change.

Product Rule

The product rule is a technique used in differentiation when dealing with the product of two functions that are both differentiable. It states that the derivative of a product of two functions is the first function times the derivative of the second plus the second function times the derivative of the first. This rule is essential for handling expressions where functions are multiplied together.

Evaluation of Derivatives at a Point

Evaluating a derivative at a specific point involves computing the derivative function and then substituting a given value of the variable into it. This process provides the instantaneous rate of change of the function at that particular point, which is crucial for understanding the local behavior of the function in a variety of applications.

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