Find dy/dx y=x^0.6
The problem involves differential calculus, specifically determining the derivative of the function y with respect to x. The function in question is y = x^0.6, where x is the independent variable and y is the dependent variable. The notation dy/dx represents the derivative of y with respect to x, which asks for the rate at which y changes as x changes incrementally. The task requires applying differentiation rules to the given power function to find the expression that represents this rate of change.
$y = x^{0.6}$
Answer
Solution:
Step 1:
Apply the differentiation operator to both sides of the equation: $\frac{d}{dx}(y) = \frac{d}{dx}(x^{0.6})$.
Step 2:
The derivative of $y$ in terms of $x$ is denoted as $\frac{dy}{dx}$.
Step 3:
Proceed to differentiate the power function on the right-hand side.
Step 3.1:
Utilize the Power Rule for differentiation, which asserts that $\frac{d}{dx}(x^n) = nx^{n-1}$ for a given $n = 0.6$. This yields $0.6x^{0.6 - 1}$.
Step 3.2:
Proceed to simplify the resulting expression.
Step 3.2.1:
Apply the rule for negative exponents: $a^{-m} = \frac{1}{a^m}$. This transforms $0.6x^{-0.4}$ into $0.6 \cdot \frac{1}{x^{0.4}}$.
Step 3.2.2:
Combine the constant with the reciprocal power to form $\frac{0.6}{x^{0.4}}$.
Step 4:
Express the derivative of $y$ with respect to $x$ by equating the left-hand side to the simplified right-hand side: $\frac{dy}{dx} = \frac{0.6}{x^{0.4}}$.
Knowledge Notes:
The problem involves finding the derivative of a function with respect to $x$, which is a fundamental concept in calculus known as differentiation. The specific steps taken in the solution are based on the following knowledge points:
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 $y$ in terms of the variable $x$.
Power Rule: A basic rule of differentiation that states if $f(x) = x^n$, then $f'(x) = nx^{n-1}$, where $n$ is a real number. This rule is used to differentiate monomial terms.
Negative Exponent Rule: A rule in algebra that states $a^{-m} = \frac{1}{a^m}$, which is used to simplify expressions with negative exponents.
Simplification: The process of rewriting an expression in a simpler or more concise form. In the context of differentiation, this often involves applying algebraic rules to make the derivative easier to understand or compute.
By understanding and applying these principles, one can solve a variety of problems involving the differentiation of algebraic functions.
