Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
s(t) = 4√t - 1/4t⁴+t+1

Verified step by step guidance

1. Step 1: Identify the function s(t) = 4√t - 1/4t⁴ + t + 1. We need to find the derivative of this function with respect to t.
2. Step 2: Break down the function into individual terms: 4√t, -1/4t⁴, t, and 1. We will find the derivative of each term separately.
3. Step 3: For the term 4√t, rewrite it as 4t^(1/2). Use the power rule for derivatives, which states that d/dx [x^n] = n*x^(n-1). Apply this rule to find the derivative of 4t^(1/2).
4. Step 4: For the term -1/4t⁴, apply the power rule again. The derivative of -1/4t⁴ is found by multiplying the exponent by the coefficient and reducing the exponent by one.
5. Step 5: The derivative of t is straightforward, as it is simply 1. The derivative of a constant, such as 1, is 0. Combine the derivatives of all terms to find the derivative of the entire function s(t).

Key Concepts

Derivatives

A derivative represents the rate at which a function changes at a given point. It is a fundamental concept in calculus that measures how a function's output value changes as its input value changes. The derivative can be interpreted as the slope of the tangent line to the curve of the function at a specific point.

Power Rule

The Power Rule is a basic differentiation rule used to find the derivative of functions in the form of x^n, where n is a real number. According to this rule, the derivative of x^n is n*x^(n-1). This rule simplifies the process of differentiation for polynomial functions and is essential for solving problems involving powers of variables.

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 g(x) that is composed with another function f(x), the derivative of the composition is the derivative of f evaluated at g(x) multiplied by the derivative of g. This rule is crucial when differentiating functions that involve nested expressions, such as square roots or trigonometric functions.