Question

In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.126. eʸ = y^(ln x)

Accepted Answer

Start with the given equation: $$e^{y}$$ = $$y^{\ln x}. $$Our goal is to find $$\frac{dy}{dx}$$, the derivative of $$y$$ with respect to $$x. $$Take the natural logarithm of both sides to simplify the expression and make differentiation easier: $$\ln(e$$^{y}$$) = \ln(y$$^{\ln x}$$). $$Use logarithm properties to rewrite both sides: the left side becomes $$y ($$since $$\ln(e$$^{y}$$) = y$$), and the right side becomes $$(\ln x) \cdot \ln y ($$since $$\ln(a$$^{b}$$) = b \ln a). So$$, we have $$y = (\ln x)(\ln y). $$Differentiate both sides implicitly with respect to $$x. $$Remember that $$y is a $$function of $$x$$, so apply the chain rule when differentiating terms involving $$y. $$For the left side, $$\frac{d}{dx}[y] = \frac{dy}{dx}. $$For the right side, use the product rule: $$\frac{d}{dx}[(\ln x)(\ln y)] = \frac{1}{x} \ln y + (\ln x) \cdot \frac{1}{y} \frac{dy}{dx}. $$After differentiating, collect all terms involving $$\frac{dy}{dx} on $$one side and factor it out. Then solve for $$\frac{dy}{dx}$$ algebraically to express the derivative explicitly in terms of $$x$$ and $$y.$$

In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
126. eʸ = y^(ln x)

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Concetti chiave

Logarithmic Differentiation

Implicit Differentiation

Properties of Logarithms and Exponents