Difference between revisions of "Mathematics/Calculus/Corner cases"
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< Mathematics | Calculus
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= Derivatives = | = Derivatives = | ||
== The derivative of <math>\frac{d}{dx} x^x</math> == | |||
* Let <math>y = x^x</math>. | |||
* Take <math>\ln</math> of both sides: <math>\ln y = x \ln x</math>. | |||
* Differentiate both sides: <math>\frac{d}{dx} \ln y = \frac{d}{dx} x \ln x</math>. | |||
* Apply the chain rule on the left-hand side: <math>\frac{d}{dx} \ln y = \frac{1}{y} \cdot \frac{dy}{dx}</math>. | |||
* Apply the product rule on the right-hand side: <math>\frac{d}{dx} x \ln x = 1 \cdot \ln x + x \cdot \frac{1}{x} = \ln x + 1</math>. | |||
* Putting it together, we have <math>\frac{1}{y} \cdot \frac{dy}{dx} = \ln x + 1</math>. | |||
* Hence <math>\frac{dy}{dx} = y (\ln x + 1) = x^x (\ln x + 1)</math>. | |||
= Integrals = | = Integrals = | ||
= Limits = | = Limits = | ||
Revision as of 08:57, 22 December 2020
Derivatives
The derivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx} x^x}
- Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = x^x} .
- Take Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ln} of both sides: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ln y = x \ln x} .
- Differentiate both sides: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx} \ln y = \frac{d}{dx} x \ln x} .
- Apply the chain rule on the left-hand side: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx} \ln y = \frac{1}{y} \cdot \frac{dy}{dx}} .
- Apply the product rule on the right-hand side: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx} x \ln x = 1 \cdot \ln x + x \cdot \frac{1}{x} = \ln x + 1} .
- Putting it together, we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{y} \cdot \frac{dy}{dx} = \ln x + 1} .
- Hence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dy}{dx} = y (\ln x + 1) = x^x (\ln x + 1)} .
