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Questions and Answers
What is the derivative of $f(x) = 3x^4 - 8x^2 + 5$?
What is the derivative of $f(x) = 3x^4 - 8x^2 + 5$?
- $f'(x) = 12x^3 - 8x$
- $f'(x) = 12x^3 - 16$
- $f'(x) = 12x^3 - 16x$ (correct)
- $f'(x) = 12x^3 - 8$
What is the derivative of $h(x) = e^{2x} + rac{1}{x^2}$?
What is the derivative of $h(x) = e^{2x} + rac{1}{x^2}$?
- $h'(x) = 2e^{2x} + rac{2}{x^3}$
- $h'(x) = 2e^{2x} - rac{2}{x^3}$
- $h'(x) = 2e^{2x} + rac{-2}{x^3}$ (correct)
- $h'(x) = 2e^{2x} - rac{-2}{x^3}$
What is the derivative of $g(x) = rac{2}{x} + 4x^2$?
What is the derivative of $g(x) = rac{2}{x} + 4x^2$?
- $g'(x) = -rac{2}{x^2} + 8x$ (correct)
- $g'(x) = -rac{2}{x} + 8$
- $g'(x) = -rac{2}{x^2} + 8$
- $g'(x) = -rac{2}{x} + 8x$
What is the inverse function of $y = ext{log}_a(x)$?
What is the inverse function of $y = ext{log}_a(x)$?
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What is the value of $ ext{log}_2(8)$?
What is the value of $ ext{log}_2(8)$?
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What is the domain of the function $f(x) = ext{log}_3(x)$?
What is the domain of the function $f(x) = ext{log}_3(x)$?
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Study Notes
Derivatives
- The derivative of $f(x) = 3x^4 - 8x^2 + 5$ can be found using the power rule and sum rule of differentiation.
- The derivative of $h(x) = e^{2x} + \frac{1}{x^2}$ involves using the chain rule for the exponential function and the power rule for the rational function.
- The derivative of $g(x) = \frac{2}{x} + 4x^2$ requires applying the power rule for differentiation to the polynomial term and the reciprocal rule to the rational term.
Inverse and Logarithmic Functions
- The inverse function of $y = \log_a(x)$ is $x = a^y$.
- The value of $\log_2(8)$ is 3, since $2^3 = 8$.
- The domain of the function $f(x) = \log_3(x)$ is $(0, \infty)$, since logarithmic functions are only defined for positive real numbers.
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