Calculate the derivative of the following functions.
y = (1 + 2 tan u)^4.5

Verified step by step guidance

1. Step 1: Identify the function y = (1 + 2 \tan u)^{4.5} as a composite function, which requires the use of the chain rule to differentiate.
2. Step 2: Apply the chain rule. The chain rule states that if you have a composite function y = f(g(u)), then the derivative y' is f'(g(u)) * g'(u).
3. Step 3: Differentiate the outer function f(v) = v^{4.5} with respect to v, which gives f'(v) = 4.5v^{3.5}.
4. Step 4: Differentiate the inner function g(u) = 1 + 2 \tan u with respect to u, which gives g'(u) = 2 \sec^2 u.
5. Step 5: Combine the results from Steps 3 and 4 using the chain rule: y' = 4.5(1 + 2 \tan u)^{3.5} \cdot 2 \sec^2 u.

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 represents the slope of the tangent line to the curve of the function at any given point. Derivatives are used to find rates of change 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 a composite function. If a function y is defined as a function of u, which is itself a function of x, the chain rule states that the derivative of y with respect to x is the product of the derivative of y with respect to u and the derivative of u with respect to x. This is essential for differentiating functions like y = (1 + 2 tan u)^(4.5), where the inner function (1 + 2 tan u) is raised to a power.

Power Rule

The power rule is a basic rule for finding the derivative of a function in the form of y = x^n, where n is a real number. According to this rule, the derivative is given by dy/dx = n*x^(n-1). This rule simplifies the process of differentiation, especially when dealing with polynomial functions or functions raised to a power, making it a crucial tool in calculus.