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Compute the derivative of the function

f(x) = x^2 + 3x + 5

May 13, 2024

Compute the derivative of the function

f(x) = x^2 + 3x + 5

Generated Graph

Solution by Steps

step 1

To find the derivative of the function

f(x) = x^2 + 3x + 5

, we will apply the power rule to each term

step 2

The derivative of

x^2

2x

because the power rule states that

d/dx(x^n) = nx^{n-1}

step 3

The derivative of

3x

3

because the derivative of a linear term

ax

is the coefficient

a

step 4

The derivative of the constant

5

0

because the derivative of any constant is zero

step 5

Combining the derivatives from steps 2, 3, and 4, we get the derivative of the function

f(x)

2x + 3 + 0

step 6

Simplifying the expression, we find that the derivative of

f(x)

2x + 3

Answer

The derivative of the function

f(x) = x^2 + 3x + 5

2x + 3

Key Concept

Power Rule for Differentiation

Explanation

The power rule is used to differentiate terms of the form

x^n

, resulting in

nx^{n-1}

. For a constant term, the derivative is zero.

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Solution by Steps

step 2

The scientific community widely agrees that increased levels of carbon dioxide in the atmosphere are a significant factor contributing to global warming and climate change

step 3

This increase in carbon dioxide leads to an enhanced greenhouse effect, trapping more heat in the Earth's atmosphere and causing temperatures to rise

[21] Answer

Key Concept

Greenhouse Effect

Explanation

Increased levels of carbon dioxide in the atmosphere enhance the greenhouse effect, leading to global warming.

step 1

To understand how increased carbon dioxide leads to more extreme flooding events, we must consider the properties of air and water vapor

step 2

Warmer air resulting from increased carbon dioxide can hold more water vapor before it condenses and precipitates as rain

step 3

This increased capacity for water vapor can lead to heavier rainfall when the air cools and the water vapor condenses, potentially causing more extreme flooding events

[22] Answer

Key Concept

Capacity of Warmer Air to Hold Water Vapor

Explanation

Warmer air can hold more water vapor, which can lead to heavier rainfall and more extreme flooding events when it condenses.