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Chapter 1: Problem 2

Find the first derivatives. \(f(P)=P^{3}+3 P^{2}-7 P+2\)

Short Answer

Expert verified
The first derivative is \ f'(P) = 3P^2 + 6P - 7.

Step by step solution

01

Differentiate Each Term

The function is given as \(f(P) = P^3 + 3P^2 - 7P + 2\). To find the first derivative, differentiate each term separately using standard differentiation rules.
02

Apply Power Rule

For the term \(P^3\), using the power rule of differentiation: \((d/dP)P^n = nP^{n-1}\). This gives \((d/dP)P^3 = 3P^2\).
03

Differentiate Remaining Terms

For the term \(3P^2\), applying the power rule gives \((d/dP)3P^2 = 6P\). For the term \(-7P,\) the derivative is \((d/dP)-7P = -7\). The constant term \(+2\) differentiates to 0.
04

Combine All Derivatives

Combine the derivatives obtained in Steps 2 and 3: \[f'(P) = 3P^2 + 6P - 7\]

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Key Concepts

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

Differentiation
Differentiation is a fundamental concept in calculus. It involves finding the rate at which a function changes at any point. This rate of change is known as the derivative. When we differentiate a function, we are looking for its derivative. For example, to differentiate a polynomial function, we apply various rules to find how each term responds to changes in the variable. Differentiation helps in understanding the behavior of functions and is widely used in many fields including physics and engineering.
Power Rule
The power rule is a simple and essential rule used in differentiation. It states that to differentiate a term of the form P^n, you bring down the exponent as a coefficient and then reduce the exponent by one. Mathematically, this is represented as: (d/dP)P^n = nP^{n-1}. For example: - The term P^3 becomes 3P^2 - The term 3P^2 becomes 6P Using the power rule, differentiating polynomial functions becomes a straightforward process.
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. It has two main branches: differential calculus and integral calculus. Differential calculus, which deals with differentiation, is about understanding rates of change and slopes of curves. Integral calculus is primarily concerned with accumulation of quantities and areas under curves. Calculus helps in solving complex problems in fields such as science, economics, and engineering.

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Most popular questions from this chapter

In Exercises 65-70, compute the difference quotient $$ \frac{f(x+h)-f(x)}{h} . $$ Simplify your answer as much as possible. \(f(x)=2 x^{2}\)

Use limits to compute the following derivatives. \(f^{\prime}(2)\), where \(f(x)=x^{3}\)

Compute the difference quotient $$ \frac{f(x+h)-f(x)}{h} . $$ Simplify your answer as much as possible. \(f(x)=x^{2}-7\)

Use limits to compute \(f^{\prime}(x)\). [Hint: In Exercises \(45-48\), use the rationalization trick of Example \(8 .]\) \(f(x)=x+\frac{1}{x}\)

Let \(f(p)\) be the number of cars sold when the price is \(p\) dollars per car. Interpret the statements \(f(10,000)=200,000\) and \(f^{\prime}(10,000)=-3\).
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