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

검증된 단계별 안내

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.

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.

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).

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.

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).

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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.

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