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# Derivatives Questions - All Grades

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Find the derivative. $f(x) = root[5](x^7)$
1. $f'(x) = root[5](7x^6)$
2. $f'(x) = 7/5 root[5](x^2)$
3. $f'(x) = root[4](x^6)$
4. $f'(x) = root[5](x^2)$
Differentiate. $f(x) = (4x^100)/25$
1. $f'(x) = (4x^99)/25$
2. $f'(x) = (8x^10) / 5$
3. $f'(x) = x^99 / 625$
4. $f'(x) = 16x^99$
On what intervals is the function $f(x)=3x^3+2x^2$ increasing or decreasing?
1. Increasing: $(-oo,-4/9) uu (0, oo)$, decreasing: $(-4/9, 0)$
2. Increasing: $(-oo,-4/9)$, decreasing: $(-4/9, 0)$
3. Increasing: $(-oo,0) uu (-4/9, oo)$, decreasing: $(-4/9, 0)$
4. Increasing: $(-4/9, 0)$, decreasing: $(-oo,-4/9) uu (0, oo)$
What is the second derivative of $f(x)=3x^3+2x^2$?
1. $f''(x)=18x+4$
2. $f''(x)=9x^2+4x$
3. $f''(x)=18$
4. $f''(x)=0$
On $f(x)=2x^5+x^3-3x$
1. $(-0.65,1.4)$ is a relative and absolute maximum.
2. $(0.65,-1.4)$ is a relative and absolute minimum.
3. $(-0.65,1.4)$ is a relative maximum.
4. $(0.65,-1.4)$ is a relative minimum.
5. Both A and B.
6. Both C and D.
7. None of the above.
Differentiate. $f(x) = 3x^4 + 9sqrt(x) - 7/root[3](x^4)$
1. $f'(x) = 12x^3 + 9sqrt(x) + 7/(4root[3](x^3))$
2. $f'(x) = 12x^3 + 9/2 sqrt(x) +21/(4root[3](x))$
3. $f'(x) = 12x^3 + 9/(2sqrt(x)) + 28/(3root[3](x^7))$
4. $f'(x) = 12x^3 +9 - 7/root[3](4x^3)$
What are all the values of $x$ for which the function $f$ defined by $f(x)=x^3+3x^2-9x+7$ is increasing?
1. $-3< x<1$
2. $-1< x< 1$
3. $x<-3$ and $x>1$
4. $x<-1$ and $x>3$
5. All real numbers
What is the derivative of $f(x)=3x^3+2x^2$?
1. $f'(x)=9x^2+4x$
2. $f'(x)=3x^2+2x$
3. $f'(x)=0$
4. $f'(x)=12x^2+6x$
Identify the intervals on which the given function is increasing and decreasing.
$f(x)=x^3+x^2-x$
1. Increasing: $(-oo, -1) uu (1/3,oo)$; decreasing: $(-1, 1/3)$
2. Increasing: $(-oo, -1)$; decreasing: $(-1, 1/3)$
3. Increasing for all x
4. Increasing: $(-1,1/3)$; decreasing: $(-oo, -1) uu (1/3,oo)$
$f(x)=2x^5+x^3-3x$ is
1. increasing on $(-oo,oo)$
2. decreasing on $(-oo,-0.65)cup(0.65,oo)$ and increasing on $(-0.65,0.65)$
3. increasing on $(-oo,-0.65)cup(0.65,oo)$ and decreasing on $(-0.65,0.65)$
4. decreasing on $(-oo,-1.4)cup(1.4,oo)$ and decreasing on $(1.4,-1.4)$
What are the critical numbers of the following equation?
$f(x)=3x^3+2x^2$
1. $x=-4/9, 0$
2. $x=4/9, 0$
3. $x=-9/4, 0$
4. $x=9/4, 0$
Find the intervals of increase and decrease. $f(x) = e^x (3x^2 + 2x - 5)$
1. Increase: $(-oo,-3) uu (0,1/3); \ \$ Decrease: $(-3,0) uu (1/3,oo)$
2. Increase: $(-oo,-3) uu (1/3,oo); \ \$ Decrease: $(-3,1/3)$
3. Increase: $(-3,0) uu (1/3,oo); \ \$ Decrease: $(-oo,-3) uu (0,1/3)$
4. Increase: $(-3,1/3); \ \$ Decrease: $(-oo,-3) uu (1/3,oo)$
Find the intervals of increase and decrease for the following function. $f(x) = sqrt(3x^2 - 9x + 6)$
1. Increasing on $(3/2,oo)$ and decreasing on $(-oo, 3/2)$
2. Increasing on $(2,oo)$ and decreasing on $(-oo,1)$
3. Increasing on $(3/2,oo)$
4. Increasing on $(2,oo)$
On what intervals is the function $f(x)=3x^3+2x^2$ concave upwards and downwards?
1. concave upward $(-2/9, oo)$, concave downward $(-oo, -2/9)$
2. concave upward $(-oo, -2/9)$, concave downward $(-2/9, -oo)$
3. concave upward $(0, oo)$, concave downward $(-oo, 0)$
4. concave upward $(-oo, 0)$, concave downward $(0, oo)$
Identify the intervals on which the given function is increasing and decreasing.
$f(x)=x^2 e^(-x^2)$
1. Increasing: $(-oo, -1) uu (0,1)$; decreasing: $(-1, 0) uu (1,oo)$
2. Increasing: $(-oo, 0)$; decreasing: $(0, oo)$
3. Increasing for all x
4. Increasing: $(-1,0) uu (1,oo)$; decreasing: $(-oo, -1) uu (0,1)$
Determine the intervals of increase and decrease. $f(x) = x^2/(x^2-1)$
1. Increasing on $(-oo,0)$ and decreasing on $(0,oo)$
2. Increasing on $(-sqrt(2),0) uu (sqrt(2),oo)$ and decreasing on $(-oo,-sqrt(2)) uu (0,sqrt(2))$
3. Increasing on $(-oo,-1) uu (1,oo)$ and decreasing on $(-1,1)$
4. Increasing on $(-oo,-1) uu (-1,0)$ and decreasing on $(0,1) uu (1,oo)$