DERIVATIVE OF ALGEBRAIC FUNCTIONS — 40 Questions
40 multiple-choice questions on derivatives of algebraic functions.
Try to solve each question independently before checking the answer key.
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Problem 1
The first derivative of the function $f(x)=\left( 4x^{2}-12x \right)\left( x+2 \right)$ is...
Problem 2
Given $f(x)=ax^{2}+2x+4$ and $g(x)=x^{2}+ax-2$. If $h(x)=\dfrac{f(x)}{g(x)}$ with $h'(0)=1$, then the value of $a$ is...
Problem 3
Given $f(x)=ax^{2}-4x+1$ and $g(x)=3x^{2}+ax+2$. If $h(x)=f(x)+g(x)$ and $k(x)=f(x)g(x)$ with $h'(0)=-3$, then the value of $k'(0)$ is...
Problem 4
If $f(x)=\dfrac{1}{x^{2}}-\dfrac{1}{x}+1$, then $f' \left( \dfrac{1}{2} \right)=\cdots$
Problem 5
Given $g(x)=3-x$ and $f(x)=6x^{2}+3x-9$. If $h(x)=f(x) \cdot g(x)$, the first derivative of $h(x)$ is $h'(x)=\cdots$
Problem 6
The first derivative of the function $f(x)=\left( x-1 \right)^{2} \left( x+1 \right)$ is $f'(x)=\cdots$
Problem 7
The first derivative of $f(x)=\dfrac{x^{2}-7}{x\sqrt{x}}$ is...
Problem 8
The first derivative of $h(x)=(-x+1)^{3}$ is...
Problem 9
The first derivative of the function $y=\dfrac{2}{\sqrt{\left( 3x^{2}+5 \right)^{3}}}$ is $y'=\cdots$
Problem 10
Given $f(0)=1$ and $f'(0)=2$. If $g(x)=\dfrac{1}{\left( 2f(x)-1 \right)^{3}}$, then $g'(0)=\cdots$
Problem 11
If $f(x)=\dfrac{bx-a}{x+b}$, satisfies $f \left( 1 \right)=1$ and $f' \left( 1 \right)=2$, then $f \left( 2 \right)=\cdots$
Problem 12
If $f(x)= \dfrac{ax+b}{x^{2}+1}$ with $f(0)=f'(0)$ and $f'(-1)=1$, then $a+b=\cdots$
Problem 13
If $m$ and $n$ are real numbers and the function $f(x)=mx^{3}+2x^{2}-nx+5$ satisfies $f'(1)=f'(-5)=0$, then $3m-n=\cdots$
Problem 14
If $f$ and $g$ are differentiable functions on $\mathbb{R}$ such that $\lim\limits_{h \to 0} \dfrac{f(x+h) \left(g(x)-g(x+h) \right)}{k^{2}h}=\dfrac{x-1}{k}$ and $\lim\limits_{h \to 0} \dfrac{g(x) \left(f(x)-f(x+h) \right)}{\left( k^{2}-1 \right)h}=\dfrac{x-1}{k+1}$ for $k \gt 0$, then...
(1) $\left(fg \right)'(0)=2k-1$
(2) $\left(fg \right)'(c)=(2k-1)(c-1)$
(3) $\left(fg \right)'(x+1)=(1-2k)x$
(4) $\left(fg \right)'\left(x^{2} \right)=(2k-1)(x^{2}-1)$
Problem 15
Given differentiable functions $f$ and $g$ with $f'(2)=3$ and $g'(2)=4$. If at $x=2$, the derivative of $\left(f \cdot g \right)(x)$ is $11$ and the derivative of $\left(f^{2}+g^{2} \right)(x)$ is $20$, then the derivative of $\left( \dfrac{f}{g} \right)(x)$ at $x=2$ is...
Problem 16
Let the function $f:\mathbb{R} \rightarrow \mathbb{R}$ be defined by $f \left( 2x-3 \right)=4x^{2}+2x-5$ and $f'$ is the first derivative of $f$. The result of $f' \left( 2x-3 \right)=\cdots$
Problem 17
The function $f(x)=x^{3}+3x^{2}-9x-7$ is decreasing on the interval...
Problem 18
The graph of $f(x)=2x^{3}-3x^{2}-120x+15$ is increasing for $x$ satisfying...
Problem 19
The graph of $f(x)=\dfrac{1}{6}x^{3}-3x^{2}$ is increasing for values of $x$ satisfying...
Problem 20
The graph of $f(x)=2x^{3}-3x^{2}-12x+7$ is decreasing for $x$ satisfying...
Problem 21
If the function $f$ is given by $f(x)=x\sqrt{x+1}$, then the interval where $f$ is increasing is...
Problem 22
The function $f(x)=\dfrac{x^{2}+3}{x-1}$ is decreasing for values of $x$ satisfying...
Problem 23
The function $f(x)=4x^{3}-9x^{2}-12x+1$ is decreasing for values of $x$ satisfying...
Problem 24
The graph of $y=2x^{3}-\frac{5}{2}x^{2}-6x+5$ is increasing for $x$ satisfying...
Problem 25
The graph of $f(x)=x^{3}+\frac{3}{2}x^{2}-18x+5$ is increasing on the interval...
Problem 26
If the curve $y= \left(x^{2}-a \right)\left( 2x+b \right)^{3}$ is decreasing on the interval $-1 \lt x \lt \frac{2}{5}$, then the value of $ab=\cdots$
Problem 27
The condition for the function $f(x)=-x^{3}+\frac{1}{2}ax^{2}-\frac{1}{2}x^{2}-3x+8$ to be always decreasing for all real values of $x$ is...
Problem 28
The range of values of $p$ such that the function $f(x)=-\frac{1}{3}x^{3}+px^{2}+2px+5$ is always decreasing for all real values of $x$ is...
Problem 29
On the interval $-1 \leq x \leq 2$, the function $y=x^{3}-3x^{2}+3$ has maximum value...
Problem 30
On the interval $0 \leq x \leq 4$, the maximum distance of the curve $f(x)=x^{3}-6x^{2}+9x$ from the $x$-axis is...
Problem 31
If the function $f(x)=x^{3}+3x^{2}-9x$ on the interval $-4 \leq x \leq -1$ has maximum value $a$ and minimum value $b$, then $a+b=\cdots$
Problem 32
If the function $f(x)=x \left( 12-2x \right)^{2}$ has maximum value $p$ and minimum value $q$, then $p-q=\cdots$
Problem 33
If the function $f(x)=2x^{3}-9x^{2}+1$ attains its maximum at point $A$, then the abscissa of point $A$ is...
Problem 34
The function $f(x)= x^{4}-2x^{2}+ax+a$ has minimum value $b$ at $x=1$. The value of $a+b$ is...
Problem 35
The minimum value of $y=x^{4}-6x^{2}-3$ is...
Problem 36
The function $f(x)=x^{3}-3x^{2}-9x+5$ attains...
Problem 37
Let $f(x)=a\sqrt{x}+\dfrac{b}{\sqrt{x}}$ have an inflection point at $(4,13)$. The value of $a+b=\cdots$
Problem 38
If $x_{1}$ and $x_{2}$ are the roots of $2x^{2}-(2c-1)x-c^{3}+4=0$, then the maximum value of $x_{1}^{2}+x_{2}^{2}$ is...
Problem 39
Given that $x_{1}$ and $x_{2}$ are the roots of $x^{2}+5ax+a^{3}-4a+1=0$. The value of $a$ that maximizes $x_{1}+x_{1}x_{2}+x_{2}$ on the interval $[-3,3]$ is...
Problem 40
If $f(x)=(x-1)^{\frac{2}{3}}$, then...
(1) $f$ is defined for $x \geq 0$
(2) $f'(2)=\frac{2}{3}$
(3) $y=\frac{2}{3}x-\frac{1}{3}$ is the tangent line at $x=2$
(4) $f$ is differentiable at every point