6 Questions
What is the derivative of $f(x) = 3x^4 - 8x^2 + 5$?
$f'(x) = 12x^3 - 16x$
What is the derivative of $h(x) = e^{2x} + rac{1}{x^2}$?
$h'(x) = 2e^{2x} + rac{-2}{x^3}$
What is the derivative of $g(x) = rac{2}{x} + 4x^2$?
$g'(x) = -rac{2}{x^2} + 8x$
What is the inverse function of $y = ext{log}_a(x)$?
$y = a^x$
What is the value of $ ext{log}_2(8)$?
3
What is the domain of the function $f(x) = ext{log}_3(x)$?
$x > 0$
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.
Test your knowledge of derivatives with these practice problems! Find the derivatives of polynomial, rational, and exponential functions to sharpen your calculus skills.
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