Suppose that the functions f and g and their derivatives with respect to x have the following values at x = 0 and x = 1.





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


f. (x¹¹ + f(x))⁻², x = 1

Verified step by step guidance

1. Step 1: Recognize that the given function is (x¹¹ + f(x))⁻². To find its derivative with respect to x, we will use the chain rule and the power rule.
2. Step 2: Let u = x¹¹ + f(x). Then the function becomes u⁻². The derivative of u⁻² with respect to u is -2u⁻³. By the chain rule, multiply this by the derivative of u with respect to x.
3. Step 3: Compute the derivative of u = x¹¹ + f(x). The derivative of x¹¹ is 11x¹⁰, and the derivative of f(x) is f'(x). So, du/dx = 11x¹⁰ + f'(x).
4. Step 4: Substitute u = x¹¹ + f(x) and du/dx = 11x¹⁰ + f'(x) into the chain rule result. The derivative of the function becomes -2(x¹¹ + f(x))⁻³ * (11x¹⁰ + f'(x)).
5. Step 5: Evaluate the derivative at x = 1. From the table, f(1) = 3 and f'(1) = -1/3. Substitute these values along with x = 1 into the expression for the derivative to simplify further.

Key Concepts

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. If you have a function h(x) = f(g(x)), the derivative h'(x) is found by multiplying the derivative of the outer function f at g(x) by the derivative of the inner function g at x. This rule is essential for differentiating expressions like (x¹¹ + f(x))⁻², where the outer function is raised to a power.

Power Rule

The power rule is a basic derivative rule used to find the derivative of functions of the form x^n, where n is a real number. According to the power rule, the derivative of x^n is n*x^(n-1). This rule is crucial when differentiating terms like x¹¹ in the given expression, as it allows us to handle polynomial components efficiently.

Derivative of Inverse Functions

When differentiating an inverse function, such as (u(x))⁻², where u(x) = x¹¹ + f(x), the derivative involves applying the chain rule and the power rule. The derivative of u(x)⁻² is -2 * u(x)⁻³ * u'(x), where u'(x) is the derivative of the inner function. This concept is vital for solving the problem, as it combines multiple differentiation techniques.