Find dy/dx y=(5x)/6

Solution:

Step 1:

Apply the differentiation operator to both sides of the equation: $\frac{d}{dx}(y)=\frac{d}{dx}\left(\frac{5x}{6}\right)$.

Step 2:

The derivative of $y$ in terms of $x$ is denoted as $\frac{dy}{dx}$.

Step 3:

Proceed to differentiate the expression on the right-hand side.

Step 3.1:

Recognize that the coefficient $\frac{5}{6}$ is a constant and can be factored out of the derivative: $\frac{5}{6}\frac{d}{dx}(x)$.

Step 3.2:

Apply the Power Rule for differentiation, which indicates that the derivative of $x^n$ is $n{x}^{n-1}$, where in this case, $n=1$: $\frac{5}{6}\cdot 1$.

Step 3.3:

Simplify the expression by multiplying $\frac{5}{6}$ by $1$: $\frac{5}{6}$.

Step 4:

Combine the results to form the complete derivative equation: $\frac{dy}{dx}=\frac{5}{6}$.

Step 5:

Substitute $\frac{dy}{dx}$ for $y$ in the final expression: $\frac{dy}{dx}=\frac{5}{6}$.

Knowledge Notes:

The process of finding the derivative of a function is called differentiation. In this problem, we are asked to find the derivative of $y$ with respect to $x$, denoted as $\frac{dy}{dx}$, for the function $y=\frac{5x}{6}$.

1. Differentiation: The process of finding the derivative, which measures how a function changes as its input changes.

2. Derivative of a Constant: The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.

3. Power Rule: A fundamental rule in differentiation which states that the derivative of $x^n$ is $n{x}^{n-1}$.

4. Simplification: After applying the differentiation rules, the expression is simplified to find the final derivative.

In this problem, we used the fact that the derivative of $y$ with respect to $x$ is $\frac{dy}{dx}$, and applied the Power Rule to the term $\frac{5x}{6}$, treating $\frac{5}{6}$ as a constant. The Power Rule tells us that the derivative of $x$ is 1, so the final derivative of the function is $\frac{5}{6}$.