Find dy/dx y=x^0.5
The given problem is asking for the derivative of the function y with respect to x, where y is defined as y = x^0.5. To find dy/dx, one would need to apply differentiation rules to calculate the rate of change of y as x changes. The power rule for differentiation is particularly relevant in this scenario, as it applies to functions where the variable is raised to a power. The problem does not ask for evaluation at a specific point but for the general derivative expression of the given function.
$y = x^{0.5}$
Answer
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 $\frac{d}{dx}[x^n] = nx^{n-1}$, 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:
Differentiation: The process of finding the derivative of a function, which represents the rate of change of the function with respect to a variable.
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 $x$.
Power Rule: A basic rule of differentiation that states if $y = x^n$, then $\frac{dy}{dx} = nx^{n-1}$. This rule is applied to functions where the exponent $n$ is a real number.
Negative Exponent Rule: In algebra, $b^{-n} = \frac{1}{b^n}$ is the rule used to rewrite expressions with negative exponents as reciprocals.
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 $\frac{0.5}{x^{0.5}}$.
