9000062402
Level:
A
Find the second derivative of \(f(x) = x^{4} - 3x^{2}\)
at the point \(x_{0} = 1\).
\(6\)
\(- 2\)
\(- 6\)
\(1\)
Derivatives of Functions
9000063101
Level:
B
Differentiate the following function.
\[
f(x) = \frac{x^{2} - 1}
{x^{2} + 1}
\]
\(f'(x) = \frac{4x}
{(x^{2}+1)^{2}} ,\ x\in \mathbb{R}\)
\(f'(x) = \frac{-4x}
{x^{2}+1},\ x\in \mathbb{R}\)
\(f'(x) = \frac{4x^{3}}
{(x^{2}+1)^{2}} ,\ x\in \mathbb{R}\)
\(f'(x) = \frac{4x}
{x^{2}+1},\ x\in \mathbb{R}\)
9000063103
Level:
B
Differentiate the following function.
\[
f(x) = \frac{x^{2} - x}
{x + 1}
\]
\(f'(x) = \frac{x^{2}+2x-1}
{(x+1)^{2}} ,\ x\neq - 1\)
\(f'(x) = 2x - 1,\ x\neq - 1\)
\(f'(x) = \frac{x^{2}+2x-1}
{(x+1)^{2}} ,\ x\neq 0\)
\(f'(x) = \frac{2x}
{(x^{2}+1)^{2}} ,\ x\neq 0\)
9000063106
Level:
B
Differentiate the following function.
\[
f(x) =\sin x\cos x
\]
\(f'(x) =\cos 2x,\ x\in \mathbb{R}\)
\(f'(x) = 1,\ x\in \mathbb{R}\)
\(f'(x) = -\cos 2x,\ x\in \mathbb{R}\)
\(f'(x) = -\sin x\cos x,\ x\in \mathbb{R}\)
9000063107
Level:
B
Differentiate the following function.
\[
f(x) =\cos x(1 +\sin x)
\]
\(f'(x) =\cos ^{2}x -\sin ^{2}x -\sin x,\ x\in \mathbb{R}\)
\(f'(x) = -\sin x\cos x,\ x\in \mathbb{R}\)
\(f'(x) =\cos x,\ x\in \mathbb{R}\)
\(f'(x) =\sin x +\sin ^{2}x -\cos ^{2}x,\ x\in \mathbb{R}\)
9000063109
Level:
B
Differentiate the following function.
\[
f(x) = 3^{x}\cdot x^{3}
\]
\(f'(x) = 3^{x}x^{2}(x\ln 3 + 3),\ x\in \mathbb{R}\)
\(f'(x) = 3^{x+1}x^{2}\ln 3,\ x\in \mathbb{R}\)
\(f'(x) = 3^{x}x^{2}(x + 3),\ x\in \mathbb{R}\)
\(f'(x) = 3^{x}x^{2}(x\ln x + 3),\ x\in \mathbb{R}^{+}\)
9000063108
Level:
B
Differentiate the following function.
\[
f(x) = x^{5}\mathrm{e}^{x}
\]
\(f'(x) = x^{4}\mathrm{e}^{x}(5 + x),\ x\in \mathbb{R}\)
\(f'(x) = 5x^{4}\mathrm{e}^{x},\ x\in \mathbb{R}\)
\(f'(x) = x^{4}\mathrm{e}^{x}(x - 5),\ x\in \mathbb{R}\)
\(f'(x) = x^{4}\mathrm{e}^{x}(5 + x^{2}),\ x\in \mathbb{R}\)
9000063102
Level:
B
Differentiate the following function.
\[
f(x) = \frac{x^{2} + 1}
{2x}
\]
\(f'(x) = \frac{x^{2}-1}
{2x^{2}} ,\ x\neq 0\)
\(f'(x) = x,\ x\neq 0\)
\(f'(x) = \frac{x-1}
{2x^{2}} ,\ x\neq 0\)
\(f'(x) = \frac{x^{2}+1}
{2x^{2}} ,\ x\neq 0\)
9000063105
Level:
B
Differentiate the following function.
\[
f(x) = \frac{\sqrt{x} - 1}
{\sqrt{x} + 1}
\]
\(f'(x) = \frac{1}
{\sqrt{x}(\sqrt{x}+1)^{2}} ,\ x > 0\)
\(f'(x) = \frac{\sqrt{x}}
{(\sqrt{x}+1)^{2}} ,\ x > 0\)
\(f'(x) = \frac{2}
{x(\sqrt{x}+1)^{2}} ,\ x > 0\)
\(f'(x) = \frac{1}
{(\sqrt{x}+1)^{2}} ,\ x > 0\)
9000062409
Level:
A
Find the first derivative of the function
\(f(x) = x^{2} + x - 6\) at each of the
\(x\)-intercepts.
\(- 5,\ 5\)
\(- 3,\ 7\)
\(- 7,\ 6\)
\(- 9,\ 6\)