Unit 3

If sin(x+y)=3x−2y, then dydx=

D: 3−cos(x+y)/2+cos(x+y)

Which of the following requires the use of implicit differentiation to find dydx ?

C: y=ln(y+x)+x2

If y=2lnx, then d4y/dx4=

B: -12/x4

The point (−2,4) lies on the curve in the xy-plane given by the equation f(x)g(y)=17−x−y, where f is a differentiable function of x and g is a differentiable function of y. Selected values of f, f′, g, and g′ are given in the table above. What is the value of dydx at the point (−2,4) ?

-3

Let f be the increasing function defined by f(x)=x3+2x2+4x+5, where f(−1)=2. If g is the inverse function of f, which of the following is a correct expression for g′(2) ?

B: g′(2)=1/f′(−1)

Which of the following expressions can be differentiated using the product rule?

B: x2tan−1x

What is the slope of the line tangent to the curve y3−xy2+x3=5 at the point (1,2) ?

B: 1/8

Which of the following methods can be used to find the derivative of y=arcsinx with respect to x ?

C: 2 and 3 only

If g(x)=lnx and f is a differentiable function of x, which of the following is equivalent to the derivative of f(g(x)) with respect to x ?

D: f′(lnx)/x

ddx(cos−1x)=

A: −1/1−x2√−11−x2

ddx(sin−1x)∣∣x=12=

B: 1/1−(12)2√

Let y=f(x) be a twice-differentiable function such that f(1)=3 and dydx=4y2+7x2−−−−−−−√. What is the value of d2ydx2 at x=1 ?

C: 55

The table above gives selected values for a differentiable and decreasing function f and its derivative. If g(x)=f−1(x) for all x, which of the following is a correct expression for g′(2) ?

C: g′(2)=1/f′(4)=−1/5

The figure above shows the graph of f′, the derivative of the function f. At which of the four indicated values of x is f′′(x) greatest?

B: B

For which of the following functions would the quotient rule be considered the best method for finding the derivative?

D: y=sin(2x+1)/2x+1

Let f be a differentiable function. If h(x)=(1+f(3x))2, which of the following gives a correct process for finding h′(x) ?

D: h′(x)=2(1+f(3x))⋅f′(3x)⋅3

The graph of the decreasing differentiable function f is shown above. Also shown is the line tangent to the graph of f at the point (4,6). Let g be the inverse of f. Which of the following statements about g′ is true?

C: g′(6)=−4/3

For which of the following functions is the chain rule an appropriate method to find the derivative with respect to x ?

D: 1 and 3 only