Math / Calculus

Question12. Find the derivative of $y=x^{\log _{3} x}$

Studdy Solution

STEP 1

1. We are given the function $y = x^{\log_{3} x}$ .
2. We need to find the derivative $\frac{dy}{dx}$ .

STEP 2

1. Rewrite the expression using properties of logarithms.
2. Differentiate the rewritten expression with respect to $x$ .
3. Simplify the derivative expression.

STEP 3

Rewrite the expression $y = x^{\log_{3} x}$ using the change of base formula for logarithms:
$\log_{3} x = \frac{\log_{b} x}{\log_{b} 3}$
For simplicity, let's use natural logarithms ( $\ln$ ):
$\log_{3} x = \frac{\ln x}{\ln 3}$
Thus, the expression becomes:
$y = x^{\frac{\ln x}{\ln 3}}$
Take the natural logarithm of both sides to simplify the exponent:
$\ln y = \ln \left( x^{\frac{\ln x}{\ln 3}} \right)$
$\ln y = \frac{\ln x}{\ln 3} \cdot \ln x$
$\ln y = \frac{(\ln x)^2}{\ln 3}$

STEP 4

Differentiate both sides with respect to $x$ .
Using implicit differentiation on the left side:
$\frac{d}{dx} (\ln y) = \frac{1}{y} \cdot \frac{dy}{dx}$
Differentiate the right side:
$\frac{d}{dx} \left( \frac{(\ln x)^2}{\ln 3} \right) = \frac{1}{\ln 3} \cdot 2\ln x \cdot \frac{1}{x}$
$= \frac{2\ln x}{x \ln 3}$
Equating both sides:
$\frac{1}{y} \cdot \frac{dy}{dx} = \frac{2\ln x}{x \ln 3}$

STEP 5

Solve for $\frac{dy}{dx}$ :
$\frac{dy}{dx} = y \cdot \frac{2\ln x}{x \ln 3}$
Substitute back the expression for $y$ :
$\frac{dy}{dx} = x^{\frac{\ln x}{\ln 3}} \cdot \frac{2\ln x}{x \ln 3}$
$\frac{dy}{dx} = x^{\frac{\ln x}{\ln 3} - 1} \cdot \frac{2\ln x}{\ln 3}$
The derivative of $y = x^{\log_{3} x}$ is:
$\frac{dy}{dx} = x^{\frac{\ln x}{\ln 3} - 1} \cdot \frac{2\ln x}{\ln 3}$

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