Calculate the derivative of the following functions.
y = cos^7/4(4x³)

Verified step by step guidance

1. Step 1: Recognize that the function y = \cos^{7/4}(4x^3) is a composition of functions, which requires the use of the chain rule for differentiation.
2. Step 2: Let u = 4x^3. Then, the function becomes y = (\cos(u))^{7/4}. Differentiate y with respect to u using the power rule: \frac{dy}{du} = \frac{7}{4}(\cos(u))^{3/4}(-\sin(u)).
3. Step 3: Differentiate u = 4x^3 with respect to x: \frac{du}{dx} = 12x^2.
4. Step 4: Apply the chain rule: \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}. Substitute the expressions from Steps 2 and 3 into this formula.
5. Step 5: Simplify the expression obtained in Step 4 to get the derivative of the original function y with respect to x.

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus that provides the slope of the tangent line to the curve at any given point. The derivative is often denoted as f'(x) or dy/dx and can be calculated using various rules, such as the power rule, product rule, and chain rule.

Chain Rule

The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function y = f(g(x)), the derivative is given by dy/dx = f'(g(x)) * g'(x). This rule is essential when differentiating functions that are nested within each other, such as trigonometric functions combined with polynomial functions.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate angles to the ratios of sides in right triangles. In calculus, these functions are important for modeling periodic phenomena and are often differentiated or integrated. Understanding their derivatives, such as the derivative of cos(x) being -sin(x), is crucial for solving problems involving trigonometric expressions.