CALCU-Compute the derivative of the function f(x)

Compute the derivative of the function

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

.

Solution by Steps

step 1

To compute the derivative of the function

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

, we will use the power rule for differentiation, which states that the derivative of

x^n

is

nx^{n-1}

step 2

Differentiating the term

x^2

using the power rule:

\frac{d}{dx}(x^2) = 2x^{2-1} = 2x

step 3

Differentiating the term

3x

using the power rule:

\frac{d}{dx}(3x) = 3x^{1-1} = 3

step 4

Differentiating the constant term

5

:

\frac{d}{dx}(5) = 0

step 5

Combining the derivatives from steps 2, 3, and 4 to find the derivative of the entire function:

f'(x) = 2x + 3 + 0

Answer

f'(x) = 2x + 3

Key Concept

Power Rule for Differentiation

Explanation

The derivative of a polynomial function is found by applying the power rule to each term separately and then combining the results.

What is the general formula for finding the derivative of a quadratic function?

Solution by Steps

step 1

Consider a general quadratic function in the form

f(x) = ax^2 + bx + c

, where

a

,

b

, and

c

are constants

step 2

Apply the power rule for differentiation, which states that the derivative of

x^n

with respect to

x

is

nx^{n-1}

step 3

Differentiate each term of the quadratic function separately: For the first term

ax^2

, the derivative is

2ax

. For the second term

bx

, the derivative is

b

. The derivative of the constant term

c

is

0

step 4

Combine the derivatives of the individual terms to obtain the derivative of the entire quadratic function

Answer

The general formula for finding the derivative of a quadratic function

f(x) = ax^2 + bx + c

is

f'(x) = 2ax + b

.

Key Concept

Power Rule for Differentiation

Explanation

The derivative of a quadratic function is found by applying the power rule to each term, resulting in a linear function where the coefficient of

x

is twice the original quadratic coefficient and the constant term is the original linear coefficient.

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