Question

Find \(d y / d x\) using logarithmic differentiation. $$ y=x^{2 / x} $$

Short answer

The derivative of the function \( y = x^{2 / x} \) is \( dy/dx = x^{2 / x} \cdot \frac{2[ \frac{1}{x} - \ln{x}]}{x^2} \).

Step-by-step solution

Apply Natural Logarithm on both sides

Apply natural logarithm on both sides of \( y = x^{2 / x} \) which gives the new equation \( \ln(y) = \ln(x^{2 / x}) \). By the properties of logarithm, this equation can be rewritten as \( \ln(y) = \frac{2 \ln(x)}{x} \).

Differentiate both sides

Next, differentiate both sides with respect to \( x \). The derivative of \( \ln(y) \) with respect to \( x \) is \( \frac{1}{y} * \frac{dy}{dx} \), and the derivative of \( \frac{2 \ln(x)}{x} \) with respect to \( x \) using the quotient rule for differentiation is \( \frac{2[ \frac{1}{x} - \ln{x}]}{x^2} \). Thus, we obtain the equation \( \frac{1}{y} * \frac{dy}{dx} = \frac{2[ \frac{1}{x} - \ln{x}]}{x^2} \). This step involves applying the chain rule and the quotient rule for differentiation.

Solve for \(dy/dx\)

Now, isolate the value of \( dy/dx \) in the equation: \( \frac{dy}{dx} = y \cdot \frac{2[ \frac{1}{x} - \ln{x}]}{x^2} \). Finally, replace \( y \) by its original expression \( x^{2 / x} \) to get the value of \( dy/dx \) without \( y \) present in it. The final expression for \( dy/dx \) is \( dy/dx = x^{2 / x} \cdot \frac{2[ \frac{1}{x} - \ln{x}]}{x^2} \).