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Chapter 1: Problem 4

Find the first derivatives. \(g(y)=y^{2}-2 y+4\)

Short Answer

Expert verified
The first derivative is \(g'(y) = 2y - 2\).

Step by step solution

01

- Understand the Problem

The task is to find the first derivative of the function \(g(y) = y^{2} - 2y + 4\). This involves using differentiation rules.
02

- Differentiate Each Term

Differentiate each term of the function separately. For \(y^2\), the derivative is \(2y\). For \(-2y\), the derivative is \(-2\). For the constant \(4\), the derivative is \(0\).
03

- Combine the Derivatives

Combine the derivatives of each term. The first derivative of \(g(y)\) is \(g'(y) = 2y - 2\).

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Key Concepts

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

Differentiation Rules
To tackle any derivative problem, you need to understand the basic rules of differentiation. The most common rules include:
  • Power Rule: For any function of the form \(y^n\), the derivative is got by bringing the exponent in front and reducing the exponent by one. Mathematically, if \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\).
  • Constant Rule: The derivative of a constant is always zero. For example, if \(f(x) = c\), where \(c\) is a constant, then \(f'(x) = 0\).
  • Sum and Difference Rule: The derivative of a sum or difference of functions is the sum or difference of their individual derivatives. For instance, if \(f(x) = g(x) \pm h(x)\), then \(f'(x) = g'(x) \pm h'(x)\).
These rules are fundamental and easy to apply once you have learnt them.
Polynomial Differentiation
Polynomial differentiation uses simple rules to find the derivatives of polynomial functions. A polynomial function is a sum of terms where each term has a variable raised to a nonnegative integer exponent and a coefficient. For the function given, \(g(y) = y^2 - 2y + 4\), it is made up of three terms:
  • The first term, \(y^2\), is a power of y.
  • The second term, \(-2y\), is a linear term.
  • The third term, \(4\), is a constant.
To differentiate a polynomial, apply the power rule to each term. Ignore the constants when differentiating, as their derivatives are zero.
First Derivative Calculation
Let's compute the first derivative of the function step-by-step using the basic differentiation rules.
Start with the function:
  • \(g(y) = y^2 - 2y + 4\)

Now, differentiate each term:
  • Differentiating the first term: Using the power rule, \(\frac{d}{dy} y^2 = 2y\).
  • Differentiating the second term: Applying the power rule and constant multiple rule, \(\frac{d}{dy} (-2y) = -2\).
  • Differentiating the constant term: According to the constant rule, \(\frac{d}{dy} 4 = 0\).

Combine the results to get the first derivative:
  • \(g'(y) = 2y - 2 + 0 = 2y - 2\).
So, the first derivative of the function \(g(y)\) is \(g'(y) = 2y - 2\). Remember to always differentiate term by term and then combine the derivatives at the end.

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Most popular questions from this chapter

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