Find dy/dx y=x^(- square root of 11)

Solution:

Step:1

Transform the radical expression $\sqrt{11}$ into an exponent by applying the rule $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$. Thus, the function becomes $y = x^{-11^{\frac{1}{2}}}$.

Step:2

Apply the differentiation operator $\frac{d}{dx}$ to both sides of the equation to obtain $\frac{d}{dx} y = \frac{d}{dx} x^{-11^{\frac{1}{2}}}$.

Step:3

The derivative of $y$ with respect to $x$ is denoted by the symbol $\frac{dy}{dx}$.

Step:4

Employ the Power Rule for differentiation, which states that $\frac{d}{dx} x^n = nx^{n-1}$, where in this case, $n = -11^{\frac{1}{2}}$. This yields $-11^{\frac{1}{2}} x^{-11^{\frac{1}{2}} - 1}$.

Step:5

Express the derivative by equating the left-hand side to the differentiated right-hand side, resulting in $\frac{dy}{dx} = -11^{\frac{1}{2}} x^{-11^{\frac{1}{2}} - 1}$.

Step:6

Substitute $\frac{dy}{dx}$ for $y$ to finalize the derivative as $\frac{dy}{dx} = -11^{\frac{1}{2}} x^{-11^{\frac{1}{2}} - 1}$.

Knowledge Notes:

The problem involves finding the derivative of a function with respect to $x$, where the function is given as $y = x^{-\sqrt{11}}$. The steps to solve this problem involve several key knowledge points:

1. Radicals as Exponents: The expression $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ is used to convert a radical into an exponent. This is useful because differentiation rules are more straightforwardly applied to exponential functions than to radical functions.

2. Differentiation: The process of finding the derivative of a function, which is the rate at which the function value changes with respect to changes in its input value.

3. Derivative Notation: $\frac{dy}{dx}$ represents the derivative of $y$ with respect to $x$, and $\frac{d}{dx}$ is the differentiation operator.

4. Power Rule: A fundamental rule in differentiation, which states that the derivative of $x^n$ with respect to $x$ is $nx^{n-1}$. This rule is applied directly to find the derivative of power functions.

5. Negative Exponents: The function involves a negative exponent, which follows the same differentiation rules as positive exponents but requires careful handling of the negative sign.

6. Simplifying Expressions: After differentiation, it is often necessary to simplify the expression to make the result clearer or to prepare it for further calculations.

By understanding and applying these concepts, one can find the derivative of the given function with respect to $x$.