Find dy/dx y=x+1/x

Solution:

Step 1:

Apply the derivative operator $\frac{d}{dx}$ to both sides of the equation $y = x + \frac{1}{x}$.

Step 2:

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

Step 3:

Proceed to differentiate the right-hand side of the equation.

Step 3.1:

Begin differentiation.

Step 3.1.1:

Utilize the Sum Rule in differentiation, which allows us to separate the derivative of a sum into the sum of the derivatives: $\frac{d}{dx}(x) + \frac{d}{dx}\left(\frac{1}{x}\right)$.

Step 3.1.2:

Apply the Power Rule for differentiation, which states that the derivative of $x^n$ is $nx^{n-1}$, to the term $x$ (where $n=1$) to obtain $1 + \frac{d}{dx}\left(\frac{1}{x}\right)$.

Step 3.2:

Focus on evaluating $\frac{d}{dx}\left(\frac{1}{x}\right)$.

Step 3.2.1:

Rewrite $\frac{1}{x}$ as $x^{-1}$ and then differentiate: $1 + \frac{d}{dx}(x^{-1})$.

Step 3.2.2:

Again, apply the Power Rule for differentiation to $x^{-1}$ (where $n=-1$), resulting in $1 - x^{-2}$.

Step 3.3:

Convert the negative exponent to a fraction using the rule $b^{-n} = \frac{1}{b^n}$ to get $1 - \frac{1}{x^2}$.

Step 3.4:

Rearrange the terms to simplify the expression: $-\frac{1}{x^2} + 1$.

Step 4:

Combine the results to form the complete derivative equation: $\frac{dy}{dx} = -\frac{1}{x^2} + 1$.

Step 5:

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

Knowledge Notes:

To solve this problem, we used several fundamental concepts of calculus:

1. Derivative: The derivative of a function measures how the function value changes as its input changes. Notationally, the derivative of $y$ with respect to $x$ is written as $\frac{dy}{dx}$.

2. Sum Rule: This rule states that the derivative of a sum of functions is the sum of the derivatives of each function. Formally, if $f(x) = g(x) + h(x)$, then $\frac{d}{dx}f(x) = \frac{d}{dx}g(x) + \frac{d}{dx}h(x)$.

3. Power Rule: The Power Rule is used to differentiate functions of the form $x^n$, where $n$ is any real number. The rule states that $\frac{d}{dx}x^n = nx^{n-1}$.

4. Negative Exponent Rule: This rule is used to simplify expressions with negative exponents. It states that $b^{-n} = \frac{1}{b^n}$, which allows us to rewrite terms with negative exponents as fractions.

By applying these rules, we can find the derivative of the given function $y = x + \frac{1}{x}$. The process involves differentiating each term separately and then combining the results to get the final derivative.