In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
47. y=(arccot(x³))³

Verified step by step guidance

1. Identify the function y = (arccot(x^3))^3 as a composition of functions, where the outer function is u^3 and the inner function is u = arccot(x^3).
2. Apply the chain rule for differentiation: \( \frac{dy}{dx} = 3 (arccot(x^3))^2 \cdot \frac{d}{dx}[arccot(x^3)] \).
3. Recall the derivative of \( arccot(z) \) with respect to z is \( -\frac{1}{1+z^2} \). Here, \( z = x^3 \), so use the chain rule again to find \( \frac{d}{dx}[arccot(x^3)] = -\frac{1}{1+(x^3)^2} \cdot \frac{d}{dx}[x^3] \).
4. Calculate \( \frac{d}{dx}[x^3] = 3x^2 \), then substitute back to get \( \frac{d}{dx}[arccot(x^3)] = -\frac{3x^2}{1+x^6} \).
5. Combine all parts to write the derivative as \( \frac{dy}{dx} = 3 (arccot(x^3))^2 \cdot \left(-\frac{3x^2}{1+x^6}\right) \), which simplifies the expression for the derivative.

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 is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. For example, if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

Derivative of Inverse Trigonometric Functions

Inverse trigonometric functions like arccot(x) have specific derivative formulas. The derivative of arccot(x) with respect to x is -1/(1 + x²). Knowing these derivatives is essential when differentiating expressions involving inverse trig functions.

Power Rule

The power rule is used to differentiate functions of the form y = [u(x)]^n, where n is a constant. It states that the derivative is n times the function raised to the (n-1) power multiplied by the derivative of the inner function u(x). This rule is often combined with the chain rule for composite functions.