Solution: Step 1: Apply the differentiation operator to both sides of the equation: fracddx(y) = fracddxleft(frac3x8right). Step 2: The derivative of y with respect to x is denoted as fracdydx. Step 3: Proceed to differentiate the expression on the right side. Step 3.1: Recognize that frac38 is a constant factor and can be pulled out of the derivative: frac38fracddx(x). Step 3.2: Apply the Power Rule for differentiation, which asserts that the derivative of xn is nxn-1, where in this case, n=1: frac38 cdot 1. Step 3.3: Simplify by multiplying frac38 with 1: frac38. Step 4: Combine the results to form the complete derivative equation: fracdydx = frac38. Step 5: Substitute fracdydx for y in the derivative equation: fracdydx = frac38. Knowledge Notes: 1. Differentiation is a fundamental concept in calculus that measures how a function changes as its input changes. The derivative of a function at a point is the slope of the tangent line to the function at that point. 2. The notation fracddx represents the differentiation operator with respect to the variable x. 3. The Power Rule is a basic differentiation rule that states if f(x) = xn, then f'(x) = nxn-1, where n is a real number. 4. Constants, such as frac38 in this problem, are unaffected by the differentiation operator and can be factored out. 5. When differentiating a simple linear function such as f(x) = x, the result is 1 since the slope of the function is constant and equal to 1. 6. After differentiating both sides of an equation with respect to x, it's important to correctly notate the derivative of y with respect to x as fracdydx.

Find dy/dx y=(3x)/8

The question provided is asking for the derivative of the function y with respect to the variable x, where the function y is defined as y=(3x)/8. Specifically, it wants you to apply differentiation techniques to find the rate at which y changes as x changes, which is a fundamental concept in calculus. This process involves applying the rules of differentiation to the given algebraic expression to calculate dy/dx, which is the notation used for the derivative of y with respect to x.

$y = \frac{3 x}{8}$

Answer

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Solution:

Step 1:

Apply the differentiation operator to both sides of the equation: $\frac{d}{dx}(y) = \frac{d}{dx}\left(\frac{3x}{8}\right)$.

Step 2:

The derivative of $y$ with respect to $x$ is denoted as $\frac{dy}{dx}$.

Step 3:

Proceed to differentiate the expression on the right side.

Step 3.1:

Recognize that $\frac{3}{8}$ is a constant factor and can be pulled out of the derivative: $\frac{3}{8}\frac{d}{dx}(x)$.

Step 3.2:

Apply the Power Rule for differentiation, which asserts that the derivative of $x^n$ is $nx^{n-1}$, where in this case, $n=1$: $\frac{3}{8} \cdot 1$.

Step 3.3:

Simplify by multiplying $\frac{3}{8}$ with $1$: $\frac{3}{8}$.

Step 4:

Combine the results to form the complete derivative equation: $\frac{dy}{dx} = \frac{3}{8}$.

Step 5:

Substitute $\frac{dy}{dx}$ for $y$ in the derivative equation: $\frac{dy}{dx} = \frac{3}{8}$.

Knowledge Notes:

Differentiation is a fundamental concept in calculus that measures how a function changes as its input changes. The derivative of a function at a point is the slope of the tangent line to the function at that point.
The notation $\frac{d}{dx}$ represents the differentiation operator with respect to the variable $x$.
The Power Rule is a basic differentiation rule that states if $f(x) = x^n$, then $f'(x) = nx^{n-1}$, where $n$ is a real number.
Constants, such as $\frac{3}{8}$ in this problem, are unaffected by the differentiation operator and can be factored out.
When differentiating a simple linear function such as $f(x) = x$, the result is 1 since the slope of the function is constant and equal to 1.
After differentiating both sides of an equation with respect to $x$, it's important to correctly notate the derivative of $y$ with respect to $x$ as $\frac{dy}{dx}$.