Problem 18 Compute the derivative of the gi... [FREE SOLUTION]

Chapter 2: Problem 18

Compute the derivative of the given function. $$p(s)=\frac{1}{4} s^{4}+\frac{1}{3} s^{3}+\frac{1}{2} s^{2}+s+1$$

Short Answer

Expert verified

The derivative of $ p(s) $ is $ s^3 + s^2 + s + 1 $.

Step by step solution

Review the Function

The given function is $ p(s) = \frac{1}{4}s^4 + \frac{1}{3}s^3 + \frac{1}{2}s^2 + s + 1 $. We need to find the derivative of this polynomial function with respect to $ s $.

Apply the Power Rule

The power rule states that the derivative of $ x^n $ is $ nx^{n-1} $. We'll apply this rule to each term in the polynomial separately.

Differentiate Each Term

Differentiate each term of the polynomial function:- For $ \frac{1}{4}s^4 $, the derivative is $ \frac{1}{4} \times 4s^{3} = s^3 $.- For $ \frac{1}{3}s^3 $, the derivative is $ \frac{1}{3} \times 3s^{2} = s^2 $.- For $ \frac{1}{2}s^2 $, the derivative is $ \frac{1}{2} \times 2s^{1} = s $.- For $ s $, the derivative is $ 1 $.- For the constant $ 1 $, the derivative is $ 0 $.

Combine the Results

Combine all the derivatives of the terms:$ s^3 + s^2 + s + 1 $ to get the derivative of the entire function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Functions

A polynomial function is a mathematical expression that involves a sum of powers in one or more variables multiplied by coefficients. In essence, it consists of terms in the form of $ ax^n $, where $ a $ is a constant coefficient and $ n $ is a non-negative integer, which is known as the degree of the term. For example:

$ s^4 $ is a polynomial term where the degree is 4.
The coefficient $ \frac{1}{4} $ in front of $ s^4 $ indicates that this term contributes a quarter of the power space described by $ s^4 $.

Polynomial functions like $ p(s) = \frac{1}{4}s^4 + \frac{1}{3}s^3 + \frac{1}{2}s^2 + s + 1 $ are common in calculus. These functions are continuous and differentiable, making them significant in mathematical studies and practical applications in physics and engineering. They are used to represent complex surfaces and curves in a simple mathematical form.

Power Rule

The power rule is an essential tool in calculus for differentiating polynomial functions. It is a straightforward technique used to find the derivative of a power function, which simplifies many differentiation problems. The power rule states: if you have a term $ x^n $, its derivative with respect to $ x $ is $ nx^{n-1} $.Using the power rule, you can see how each term of a polynomial function is handled:

The exponent $ n $ of the term $ x^n $ is multiplied by the coefficient $ a $.
Then, the exponent is reduced by one to get the new term of the derivative.

For example, for the term $ \frac{1}{4}s^4 $, apply the power rule:

Multiply $ \frac{1}{4} \times 4 = 1 $.
The new term is $ s^{3} $ after subtracting 1 from the exponent.

The power rule simplifies the process of differentiation, especially when dealing with polynomials, where each term's derivative can be calculated independently, and the results are then summed.

Differentiation Steps

Differentiation is a key process in calculus that involves computing the derivative of a function. This process was applied to find the derivative of the polynomial function $ p(s) = \frac{1}{4}s^4 + \frac{1}{3}s^3 + \frac{1}{2}s^2 + s + 1 $. Here's how to differentiate it step by step:

Identify each term in the polynomial function separately.
Apply the power rule to each term, performing multiplication and subtraction for the exponent.
For the first term, $ \frac{1}{4}s^4 $, the derivative becomes $ s^3 $.
The second term, $ \frac{1}{3}s^3 $, becomes $ s^2 $.
The third term, $ \frac{1}{2}s^2 $, simply reduces to $ s $.
The fourth term, $ s $, turns into $ 1 $ since any constant multiplied by $ s^1 $ differentiates to the constant itself.
The constant term $ 1 $ turns into 0 as constants have no slope.

Combining these, the derivative of our function $ p(s) $ is $ s^3 + s^2 + s + 1 $. Each step simplifies an individual part of the function, resulting in a clear and comprehensive derivative that can be used in further analyses.

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91影视

Compute the derivative of the given function. $$p(s)=\frac{1}{4} s^{4}+\frac{1}{3} s^{3}+\frac{1}{2} s^{2}+s+1$$

Short Answer

Step by step solution

Review the Function

Apply the Power Rule

Differentiate Each Term

Combine the Results

Key Concepts

Polynomial Functions

Power Rule

Differentiation Steps

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