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Questions and Answers
What method is primarily used to differentiate the function y = (7 - 3x) / √(3(6x + 1) - (3(3x^2 + x)(-31 · x - 3)))?
What method is primarily used to differentiate the function y = (7 - 3x) / √(3(6x + 1) - (3(3x^2 + x)(-31 · x - 3)))?
- Product rule
- Quotient rule (correct)
- Sum rule
- Chain rule
For the function y = tan^4(5x^6) + Arcsec(cos x), what is the first part of the derivative of the function?
For the function y = tan^4(5x^6) + Arcsec(cos x), what is the first part of the derivative of the function?
- 24 tan^3(5x^6) sec(5x^6)
- 20x^5 tan^3(5x^6) sec^4(5x^6)
- 4 tan^4(5x^6) sec(5x^6)
- 4 tan^3(5x^6) sec^2(5x^6) * 30x^5 (correct)
In the differentiation of y = 4csc(x) log5(3x^4), what rule is correctly applied?
In the differentiation of y = 4csc(x) log5(3x^4), what rule is correctly applied?
- Quotient rule
- Sum rule
- Chain rule
- Product rule (correct)
What is the derivative of the function y = 4csc(x) log5(3x^4) in terms of its components?
What is the derivative of the function y = 4csc(x) log5(3x^4) in terms of its components?
For y = x^2 cot(x + 2y), which aspect must be considered due to y being a function of x?
For y = x^2 cot(x + 2y), which aspect must be considered due to y being a function of x?
What is the limit of the expression as x approaches 16: $\lim_{x \to 16} \frac{\sqrt{x} - 4}{x - 16}$?
What is the limit of the expression as x approaches 16: $\lim_{x \to 16} \frac{\sqrt{x} - 4}{x - 16}$?
At what value is the function $f(x)$ continuous when x equals 2?
At what value is the function $f(x)$ continuous when x equals 2?
What is the limit from the left of the function $f(x)$ as x approaches 2?
What is the limit from the left of the function $f(x)$ as x approaches 2?
What does the limit $\lim_{x \to 2} f(x)$ indicate about the function at x = 2?
What does the limit $\lim_{x \to 2} f(x)$ indicate about the function at x = 2?
If $f(x) = \cos \frac{\pi}{2}$ for $x \leq 2$, what is the value of f(2)?
If $f(x) = \cos \frac{\pi}{2}$ for $x \leq 2$, what is the value of f(2)?
What expression can be used to evaluate the limit $\lim_{x \to 16} \frac{\sqrt{x} - 4}{x - 16}$?
What expression can be used to evaluate the limit $\lim_{x \to 16} \frac{\sqrt{x} - 4}{x - 16}$?
What type of discontinuity does $f(x)$ exhibit at x = 2?
What type of discontinuity does $f(x)$ exhibit at x = 2?
If $y = \sqrt{3x^2 + x}$, what method is typically used to find the derivative, $y'$?
If $y = \sqrt{3x^2 + x}$, what method is typically used to find the derivative, $y'$?
What is the limit of the expression $\lim_{x \to +\infty} \frac{x \sin x}{x}$?
What is the limit of the expression $\lim_{x \to +\infty} \frac{x \sin x}{x}$?
Which rule is used to evaluate the limit of $\lim_{x \to +\infty} \frac{\ln |x|}{x}$?
Which rule is used to evaluate the limit of $\lim_{x \to +\infty} \frac{\ln |x|}{x}$?
What is the value of $\lim_{x \to +\infty} \cos x$?
What is the value of $\lim_{x \to +\infty} \cos x$?
What is the final result of $\lim_{x \to +\infty} x^{1/x}$?
What is the final result of $\lim_{x \to +\infty} x^{1/x}$?
In the expression $\lim_{x \to +\infty} \frac{\ln |y|}{x}$, what is the behavior of $\ln |y|$ as $x$ approaches infinity?
In the expression $\lim_{x \to +\infty} \frac{\ln |y|}{x}$, what is the behavior of $\ln |y|$ as $x$ approaches infinity?
When applying L'Hôpital's rule to $\lim_{x \to +\infty} \frac{\ln |x|}{x}$, what is the result of the derivative of $\ln |x|$?
When applying L'Hôpital's rule to $\lim_{x \to +\infty} \frac{\ln |x|}{x}$, what is the result of the derivative of $\ln |x|$?
What indeterminate form is encountered when evaluating $\lim_{x \to +\infty} \frac{x \sin x}{x}$?
What indeterminate form is encountered when evaluating $\lim_{x \to +\infty} \frac{x \sin x}{x}$?
What is the final limit of $\lim_{x \to +\infty} x^{1/x}$?
What is the final limit of $\lim_{x \to +\infty} x^{1/x}$?
What is the result of applying the product rule to the expression $Dx[y] = Dx[x^2 cot(x + 2y)]$?
What is the result of applying the product rule to the expression $Dx[y] = Dx[x^2 cot(x + 2y)]$?
What is the initial step in finding $dy/dx$ if $y = Arccot(x)$?
What is the initial step in finding $dy/dx$ if $y = Arccot(x)$?
When differentiating $y = ln(Arccot(x))$, what derivative should you expect to calculate first?
When differentiating $y = ln(Arccot(x))$, what derivative should you expect to calculate first?
In the expression $dy/dx = x sin(x) / Arccot(x)$, which part represents the function being differentiated?
In the expression $dy/dx = x sin(x) / Arccot(x)$, which part represents the function being differentiated?
What is the key factor to apply while differentiating a quotient like $x sin(x) / Arccot(x)$?
What is the key factor to apply while differentiating a quotient like $x sin(x) / Arccot(x)$?
Which trigonometric identity relates $csc^2(x + 2y)$ and $cot(x + 2y)$ in the differentiation process?
Which trigonometric identity relates $csc^2(x + 2y)$ and $cot(x + 2y)$ in the differentiation process?
What limit is being evaluated as $x$ approaches $+iginfty$ in the expression $lim_{x o +iginfty} x sin(x)$?
What limit is being evaluated as $x$ approaches $+iginfty$ in the expression $lim_{x o +iginfty} x sin(x)$?
Which of the following indicates the proper application of logarithmic properties when differentiating?
Which of the following indicates the proper application of logarithmic properties when differentiating?
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Study Notes
Limits
- The limit of a function (√x − 4)/(x − 16) as x approaches 16 is 8.
- The limit of a function f(x) is an essential discontinuity at x = 2.
- The left-hand limit of f(x) approaches 0, while the right-hand limit of f(x) at x = 2 approaches 2.
- The limit of x sin (1/x) as x approaches infinity is 1, which is found by manipulating the expression and applying L'Hopital's Rule.
- The limit of x^(1/x) as x approaches infinity is 1, which is found by manipulating the expression and applying L'Hopital's Rule.
Differentiation
- The derivative of a function (3√(3x^2 + x)) / √(7 - 3x) can be found using the quotient rule for derivatives.
- The derivative of tan^4(5x^6) + Arcsec(cos x) can be found using the chain rule for derivatives.
- The derivative of 4csc(x) log5(3x^4) can be found using the product rule for derivatives and the chain rule for derivatives.
- The derivative of x^2 cot(x + 2y), where y is a differentiable function of x, can be found using the product rule for derivatives and the chain rule for derivatives.
Logarithmic Differentiation
- The derivative of y = (x sin(x)) / arccot(x) is found using logarithmic differentiation.
L'Hopital's Rule
- L'Hopital's rule can be used to evaluate limits of functions that have an indeterminate form of 0/0 or ∞/∞.
- The derivative of x^(1/x) is found using L'Hopital's rule by manipulating the function into a form for which L'Hopital's rule is applicable.
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