66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
66. Let f(x) = cos(x²).
b. Calculate f''(x).

Verified step by step guidance

1. Recall that the function is given by \(f(x) = \cos(x^2)\). Our goal is to find the second derivative \(f''(x)\).
2. First, find the first derivative \(f'(x)\) by applying the chain rule. The derivative of \(\cos(u)\) with respect to \(u\) is \(-\sin(u)\), and here \(u = x^2\). So, \(f'(x) = -\sin(x^2) \cdot \frac{d}{dx}(x^2)\).
3. Calculate the derivative of the inner function: \(\frac{d}{dx}(x^2) = 2x\). Substitute this back to get \(f'(x) = -2x \sin(x^2)\).
4. Next, find the second derivative \(f''(x)\) by differentiating \(f'(x) = -2x \sin(x^2)\). Use the product rule: if \(h(x) = u(x)v(x)\), then \(h'(x) = u'(x)v(x) + u(x)v'(x)\).
5. Let \(u(x) = -2x\) and \(v(x) = \sin(x^2)\). Then, \(u'(x) = -2\) and \(v'(x)\) requires the chain rule again: \(v'(x) = \cos(x^2) \cdot 2x\). Substitute these into the product rule formula to express \(f''(x)\).

Key Concepts

Second Derivative

The second derivative of a function measures the rate of change of the first derivative, providing information about the function's concavity and acceleration. It is found by differentiating the first derivative once more with respect to the variable.

Chain Rule

The chain rule is a differentiation technique used when dealing with composite functions. It states that the derivative of f(g(x)) is f'(g(x)) multiplied by g'(x), allowing us to differentiate functions like cos(x²) effectively.

Derivative of Trigonometric Functions

Understanding the derivatives of basic trigonometric functions, such as d/dx[cos(x)] = -sin(x), is essential. This knowledge helps in differentiating more complex functions involving trigonometric expressions.