Derivatives Review

If y= lnx/x, then dy/dx=

(1-lnx)/x^2

d/dx (2^x)=

(2^x)ln2

The table above gives values of the differentiable functions f and g and their derivatives at x = 0. If h(x)=f(x)g(x), what is the value of h'(0) ?

(8-3pi)/4

Let f be a differentiable function such that f(3)=15, f(6)= 3, f'(3)= -8, and f'(6)= -2. The function g is differentiable and g(x)= f^-1(x) for all x. What is the value of g′ (3)?

-1/2

The graphs of the linear functions f and g are shown above. If h(x)=f(x)+g(x), then h′(x)=

-1/2

If f(x)=arccos(x^2), then f′(x)=

-2x/(√1-x^4)

What is the slope of the line tangent to the graph of y= e^-x/(x+1) at x=1?

-3/4e

In the xy-plane, what is the slope of the line tangent to the graph of x^2+xy+y^2=7 at the point (2, 1) ?

-5/4

Let f and g be differentiable functions such that f′(1)=2 and g′(1)=6. If h(x)=5f(x)−4g(x)+3x^2 −2, what is the value of h′(1) ?

-8

If f(x)=(x−1)^2 sinx, then f'(0)=

1

If f(x)=√x + (3/√x), then f'(4) =

1/16

The table above gives values of the differentiable functions f and g, and f′, the derivative of f, at selected values of x. If g (x)=f−1 (x) , what is the value of g′ (4)?

1/4

The function h is given by h (x)=x^5 + 3x−2 and h(1)=2. If h^−1 is the inverse of h, what is the value of (h^−1)′ (2)?

1/8

The graphs of the piecewise linear functions f and g are shown above. If the function h is defined by h(x) = f(x)g(x), then h'(2) is

14

If f(x)=ln(e^2x) , then f'(x)=

2

What is the slope of the line tangent to the curve y=arctan(4x) at the point at which x=14 ?

2

If y=1/2x^(4/5) −3/x^5, then dy/dx=

2/5x^(1/5)+15/x^6

If dy/dx= x^4−2x^3+3x−1 , then d3y/dx3 evaluated at x=2 is

24

If y=x^2 sin2x, then dy/dx=

2x(sin2x + xcos2x)

If f(x) =x^(3/2) then f'(4) =

3

If f(x) = e^2x(x^3 +1), then f'(2)=

30e^4

If y=sin(3x), then dy/dx=

3cos(3x)

The table above gives values of the differentiable functions f and g and their derivatives at x = 1. If h(x) = (2 f (x) + 3)(1 + g(x)), then h'(1)=

44

If y=sin^−1(5x), then dy/dx=

5/(√1-25x^2)

The table above gives values of the differentiable functions f and g and of their derivatives f′ and g′ , at selected values of x. If h(x) = f(g(x)), what is the slope of the graph of h at x = 2?

6

If f(x)=7x-3+lnx, then f'(1)=

8

If f(x)=x^3−x^2+x−1, then f′(2)=

9

d/dx (cosxtanx)=

cosx

The functions f and g are differentiable. For all x, f (g(x)) = x and g(f (x)) = x. If f (3) = 8 and f'(3)= 9, what are the values of g(8) and g'(8)?

g(8)= 3 and g'(8)= 1/9

If g is the function defined above, then g′(1) is

nonexistent

If y=tanx−cotx, then dy/dx =

sec^2 x + csc^2 x

If y=x^2e^x, then dy/dx=

xe^x(x+2)

Which of the following is an equation of the line tangent to the graph of x^2−3xy=10 at the point (1, − 3)?

y+3=11/3(x-1)

An equation of the line tangent to the graph of f(x)=x(1−2x)^3 at the point (1,−1) is

y= -7x+6