Find dy/dx y=(5x)/6
The problem presented is a calculus problem requiring differentiation. You are asked to compute the first derivative of the function y with respect to x, where y is given as a function of x in the form (5x)/6. The notation dy/dx represents the derivative, which is a measure of how y changes with a small change in x. Solving this problem involves applying differentiation rules to the algebraic expression (5x)/6.
$y = \frac{5 x}{6}$
Answer
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 $nx^{n-1}$, 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 $y = \frac{5x}{6}$.
Differentiation: The process of finding the derivative, which measures how a function changes as its input changes.
Derivative of a Constant: The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.
Power Rule: A fundamental rule in differentiation which states that the derivative of $x^n$ is $nx^{n-1}$.
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 $\frac{5}{6}$.
