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Derivatives Practice Quiz

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Questions and Answers

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 $h(x) = e^{2x} + rac{1}{x^2}$?

$h'(x) = 2e^{2x} + rac{2}{x^3}$

$h'(x) = 2e^{2x} - rac{2}{x^3}$

$h'(x) = 2e^{2x} + rac{-2}{x^3}$ (correct)

$h'(x) = 2e^{2x} - rac{-2}{x^3}$

What is the derivative of $g(x) = rac{2}{x} + 4x^2$?

$g'(x) = -rac{2}{x^2} + 8x$ (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 $y = ext{log}_a(x)$?

$y = a^x$ Signup and view all the answers

What is the value of $ ext{log}_2(8)$?

3 Signup and view all the answers

What is the domain of the function $f(x) = ext{log}_3(x)$?

$x > 0$ Signup and view all the answers

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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Description

Test your knowledge of derivatives with these practice problems! Find the derivatives of polynomial, rational, and exponential functions to sharpen your calculus skills.