Find dy/dx y=x^0.5

Solution:

Step 1:

Apply differentiation to each side of the equation $y=x^{0.5}$.

Step 2:

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

Step 3:

Proceed to differentiate the right-hand side.

Step 3.1:

Utilize the Power Rule for differentiation, which states $\frac{d}{dx}[x^n]=n{x}^{n-1}$, where $n$ is the exponent, in this case, $n=0.5$.

Step 3.2:

Simplify the resulting expression.

Step 3.2.1:

Express the negative exponent as a reciprocal, according to the rule $b^{-n}=\frac{1}{b^n}$.

Step 3.2.2:

Combine the constant $0.5$ with the reciprocal of $x^{0.5}$ to obtain $\frac{0.5}{x^{0.5}}$.

Step 4:

Formulate the derivative equation by equating the derivative of $y$ with the simplified expression from the right-hand side.

Step 5:

Substitute $\frac{dy}{dx}$ for $y$ to complete the differentiation, yielding $\frac{dy}{dx}=\frac{0.5}{x^{0.5}}$.

Knowledge Notes:

The process of finding $\frac{dy}{dx}$ for the function $y=x^{0.5}$ involves several key concepts in calculus and algebra:

1. Differentiation: The process of finding the derivative of a function, which represents the rate of change of the function with respect to a variable.

2. Derivative of $y$ with respect to $x$: Denoted as $\frac{dy}{dx}$, it is the notation used to represent the derivative of the function $y$ with respect to the variable $x$.

3. Power Rule: A basic rule of differentiation that states if $y=x^n$, then $\frac{dy}{dx}=n{x}^{n-1}$. This rule is applied to functions where the exponent $n$ is a real number.

4. Negative Exponent Rule: In algebra, $b^{-n}=\frac{1}{b^n}$ is the rule used to rewrite expressions with negative exponents as reciprocals.

5. Simplification: The process of rewriting an expression in a simpler or more compact form, often by combining like terms or using algebraic rules.

By applying these concepts, we can differentiate the function $y=x^{0.5}$ and find that $\frac{dy}{dx}=\frac{0.5}{x^{0.5}}$. This result tells us that the slope of the tangent line to the curve at any point $x$ is equal to $\frac{0.5}{x^{0.5}}$.