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Chapter 3: Problem 1

Find the derivatives of the functions. \(5 x^{3}+12 x^{2}-15 \)

Short Answer

Expert verified
The derivative is \(15x^2 + 24x\).

Step by step solution

01

Identify the Function

The function given is a polynomial function: \(5x^3 + 12x^2 - 15\). A polynomial function can be differentiated term by term.
02

Apply the Power Rule

The power rule states that the derivative of \(x^n\) is \(nx^{n-1}\). Apply the power rule to each term in the function separately. For \(5x^3\), the derivative is \(3 \times 5x^{3-1} = 15x^2\).For \(12x^2\), the derivative is \(2 \times 12x^{2-1} = 24x\).For \(-15\), the derivative of a constant is 0.
03

Write the Resulting Derivative

Combine the derivatives calculated for each term. The derivative of the function \(5x^3 + 12x^2 - 15\) is \(15x^2 + 24x\).

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Key Concepts

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

Power Rule
The Power Rule is a fundamental concept in calculus used for finding derivatives. It's a simple yet powerful tool helpful for differentiating polynomial functions. The rule states that if you have a term in the form of \(x^n\), the derivative is found by multiplying \(n\) (the exponent) by the coefficient of the term, then lowering the exponent by one. For example, if you have a term \(5x^3\), applying the power rule gives you the derivative:
  • Start by identifying the exponent \(n\) of the term, which is 3 in this case.
  • Multiply the coefficient, 5, by the exponent: \(3 \times 5 = 15\).
  • Reduce the exponent by 1: \(3 - 1 = 2\).
  • The derivative of \(5x^3\) is then \(15x^2\).
The Power Rule makes differentiating complex polynomial expressions straightforward by allowing you to tackle each term one at a time.
Polynomial Function
A polynomial function consists of terms that are composed of variables raised to whole number exponents and coefficients. In our exercise, the polynomial function is \(5x^3 + 12x^2 - 15\). A polynomial is expressed as:
  • Terms like \(5x^3\) and \(12x^2\) where the variable \(x\) is raised to a power.
  • The highest exponent gives the degree of the polynomial. The highest exponent in this function is 3, making it a cubic polynomial.
  • Polynomial functions are smooth and continuous curves without breaks or holes.
To differentiate a polynomial, you apply the Power Rule to each term separately. The simplicity of polynomial functions is that any polynomial can be differentiated term by term, which often makes them easier to work with than more complicated functions.
Differentiation
Differentiation is the calculus technique that entails finding the derivative of a function. The derivative measures how a function changes as its input changes. In simpler terms, it helps us understand the rate of change or the slope of a function at any given point. Here's a brief breakdown:
  • When given a function, differentiation allows you to determine how that function's output value changes concerning changes in its input value.
  • For polynomial functions, differentiation is straightforward, involving calculating new terms by adjusting the powers and coefficients of each term in the original function.
  • An important aspect of differentiation is that it handles each term of the polynomial separately, making it manageable.
In our example, differentiation takes the polynomial function \(5x^3 + 12x^2 - 15\) and transforms it into its derivative, \(15x^2 + 24x\), by applying the rules of differentiation to each component term.

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Most popular questions from this chapter

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible. \(\sqrt{x^{3}-x^{2}-(1 / x)}\)

Find the derivatives of the functions using the quotient rule. \(\frac{x^{3}}{x^{3}-5 x+10}\)

Find the derivatives of the functions. \(5\left(-3 x^{2}+5 x+1\right) \)

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible. \((x+8)^{5}\)

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible. \(\sqrt{1+x^{4}} \)
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