Suppose that functions f and g and their derivatives with respect to x have the following values at x = 2 and x = 3.





Find the derivatives with respect to x of the following combinations at the given value of x.


g. 1 / g²(x), x = 3

Verified step by step guidance

1. Step 1: Recall the derivative rule for the reciprocal of a function. If h(x) = 1 / g²(x), then h'(x) can be found using the chain rule and the power rule. Specifically, h'(x) = -2 * g(x)^(-3) * g'(x).
2. Step 2: Substitute the given value of x = 3 into the formula. From the table, g(x) = -4 and g'(x) = 5 at x = 3.
3. Step 3: Replace g(x) and g'(x) in the derivative formula. This gives h'(x) = -2 * (-4)^(-3) * 5.
4. Step 4: Simplify the expression step by step. First, calculate (-4)^(-3), which is the reciprocal of (-4)^3. Then multiply by -2 and g'(x).
5. Step 5: The simplified expression will give the derivative h'(x) at x = 3. Ensure all calculations are consistent with the rules of exponents and multiplication.

Key Concepts

Chain Rule

The chain rule is a fundamental differentiation technique used when dealing with composite functions. It states that the derivative of a composite function f(g(x)) is f'(g(x)) * g'(x). In this problem, the chain rule helps differentiate expressions involving powers of functions, such as g²(x), by considering the derivative of the outer function and multiplying it by the derivative of the inner function.

Power Rule

The power rule is a basic differentiation rule that states if f(x) = x^n, then f'(x) = n*x^(n-1). This rule is essential for finding the derivative of functions raised to a power, such as g²(x). In the context of the problem, it helps simplify the differentiation process by providing a straightforward method to handle powers of functions.

Quotient Rule

The quotient rule is used to differentiate functions that are ratios of two other functions. It states that if h(x) = f(x)/g(x), then h'(x) = (f'(x)g(x) - f(x)g'(x))/g²(x). In this problem, the quotient rule is applied to find the derivative of 1/g²(x), which involves differentiating a function in the form of a reciprocal, requiring careful application of the rule to ensure accuracy.