Derivatives
The derivative of
Question
What is ?
Solution 1
- Let .
- Take of both sides: .
- Differentiate both sides: .
- Apply the chain rule on the left-hand side: .
- Apply the product rule on the right-hand side: .
- Putting it together, we have .
- Hence .
Solution 2
- Note that , so .
- Applying the chain rule, .
- Applying the product rule, .
- Therefore .
Integrals
The integral
Question
What is ?
Solution
- We can write as .
- Consider the series expansion of :
.
- We can interchange the integration and summation (we can recognize this as a special case of the Fubini/Tonelli theorems) and write
Limits
The limit of
Question
What is ?
Solution
- Note that .
- We can further rewrite this as .
- As long as is continuous and the limit of exists at the point in question, the limit will commute with composition:
In our case, is continuous, so
- The question, then, is what is .
- As , . In this situation we can apply l'Hôpital's rule:
- Hence Failed to parse (unknown function "\tendsto"): {\displaystyle \lim_{x \tendsto 0^+} x^x = e^0 = 1}
.
The limits 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 \lim_{x \tendsto +\infty} x \sin \frac{1}{x}}
and Failed to parse (unknown function "\tendsto"): {\displaystyle \lim_{x \tendsto -\infty} x \sin \frac{1}{x}}
Question
What are 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 \lim_{x \tendsto +\infty} x \sin \frac{1}{x}}
and 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 \lim_{x \tendsto -\infty} x \sin \frac{1}{x}}
?
Solution
- Let us rewrite .
- As Failed to parse (unknown function "\tendsto"): {\displaystyle x \tendsto +\infty$, $\frac{1}{x} \tendsto 0}
and 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 x \sin \frac{1}{x} \tendsto 0}
.
- We have ``, so we can apply l'Hôpital's rule.
- Differentiating the numerator in , we obtain .
- Differentiating the denominator in , we obtain .
- Thus
Failed to parse (unknown function "\tendsto"): {\displaystyle \lim_{x \tendsto +\infty} x \sin \frac{1}{x} = \lim_{x \tendsto +\infty} \frac{\sin \frac{1}{x}}{\frac{1}{x}} = \lim_{x \tendsto +\infty} \frac{\left(\cos \frac{1}{x}\right) \left(-\frac{1}{x^2}\right)}{-\frac{1}{x^2}} = \lim_{x \tendsto +\infty} \cos \frac{1}{x} = 1. }
- Similarly we can find that 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 \lim_{x \tendsto -\infty} x \sin \frac{1}{x} = 1}
.