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Solved on Mar 24, 2024

Find the derivative of $y=6x$ . Options: A. $x$ , B. 0, C. $x^{2}$ , D. $6$ .

STEP 1

1. The expression $y=6x$ represents a linear function of $x$ .
2. The notation $y^{\prime}$ represents the derivative of $y$ with respect to $x$ .
3. Basic rules of differentiation apply, such as the derivative of a constant times a variable is the constant.

STEP 2

1. Differentiate $y=6x$ with respect to $x$ .
2. Identify the correct answer from the given options.

STEP 3

Differentiate the function $y=6x$ with respect to $x$ using the power rule.
$y^{\prime} = \frac{d}{dx}(6x)$

STEP 4

Apply the constant multiple rule, which states that the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.
$y^{\prime} = 6 \frac{d}{dx}(x)$

STEP 5

Use the fact that the derivative of $x$ with respect to $x$ is 1.
$y^{\prime} = 6 \cdot 1$

STEP 6

Simplify the expression to find the derivative.
$y^{\prime} = 6$

STEP 7

Compare the result with the given options.
The correct answer is D. 6.

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