Logarithmic differentiation Calculator

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Example

Created on 2024-06-20Asked by Luna Smith (Solvelet student)

Use logarithmic differentiation to find the derivative of

y = x^x

Solution

To find the derivative of

y = x^x

using logarithmic differentiation, follow these steps: 1. Take the natural logarithm of both sides:

\ln y = \ln (x^x)

2. Simplify the right-hand side using the logarithm power rule:

\ln y = x \ln x

3. Differentiate both sides with respect to

x

\frac{d}{dx} (\ln y) = \frac{d}{dx} (x \ln x)

\frac{1}{y} \frac{dy}{dx} = \ln x + 1

4. Solve for

\frac{dy}{dx}

\frac{dy}{dx} = y (\ln x + 1)

5. Substitute

y = x^x

back into the equation:

\frac{dy}{dx} = x^x (\ln x + 1)

Therefore, the derivative of

y = x^x

is:

\frac{dy}{dx} = x^x (\ln x + 1)

Solved on Solvelet with Basic AI Model

Some of the related questions asked by James Nguyen on Solvelet

1. Differentiate the function

y = (\ln(x))^2(x^2 + 1)

.2. Find the derivative of the function

y = \sqrt{x} \ln(x)

DefinitionLogarithm differentiation is a mathematical technique used to differentiate functions (by taking the natural logarithm of both sides) and simplifying using the known fact of logarithm differentiation, namely, differentiating the product of two terms and then taking the derivative of each term separately. This is especially good for functions in the form o y=f(x)g(x) or products and quotients of several functions. Example: In order to find d/dx(y=xx), first take the natural logarithm: ln(y)=xln(x) and then differentiate: y1dxdy=ln(x)+1. So dt/dx=xx(ln(x)+1), and dxdy=y(ln(x)+1)

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