Sample Questions for Midterm Two

1. Compute the derivative of the function

Solution By the quotient rule,

2. Compute the derivative of the function

Solution By the chain rule,

3. Suppose we had proved the product rule and the chain rule before doing the quotient rule. Show how you could derive the quotient rule using only the product rule and the chain rule.

Solution The idea is to write the quotient as a product and then use the product rule to differentiate.

To compute the derivative of the last term we have to use the chain rule:

4. Compute the derivative with respect to x of sin² x cos² x.

Solution by the product and chain rules we get

5. Compute the derivative of

Solution By the quotient rule we have

6. Find the derivative with respect to x of

Solution By standard log manipulations we have that this simplifies to

The derivative is

7. Consider the tangent line to the curve y = ln x at the point . Show that the point where the tangent line crosses the y-axis is no matter where on the curve we place .

Solution The derivative at the point x₀ gives the slope of the tangent line.

The tangent line has equation

or

The tangent line crosses the y-axis when x = 0:

8. Compute the derivative with respect to x of x^x.

Solution: The simplest method is to use logarithmic differentiation:

Differentiating both sides with respect to x gives

or

9. If , find dy/dx.

Solution: taking a derivative with respect to x on both sides of the equation we have

Solving this for we have

10. Compute the derivative of

Solution Before differentiating, we can use some of the properties of the log to simplify this expression.

Differentiating gives

11. One of the lines tangent to the graph of y = e^x passes through the origin. Where does that tangent line touch the graph of y = e^x ?

Solution Call the point where the tangent line touches the graph . Since the point is on the graph, we must have that . The derivative at that point, which give the slope of the tangent line, is also . Thus the equation of the tangent line is

The final fact to apply here is the requirement that the tangent line must pass through the origin:

This equation has a simple solution:

The tangent line we want touches the graph at and has equation

12. Show that the derivative of the function y = tan^-1 x is

Solution Starting from

tan y = x

and using implicit differentiation we have

or

To simplify the term in the denominator, we can use the trig identity

with y = tan^-1 x.

so

13. Compute the derivative of

Solution By the quotient rule the derivative is

14. Compute the derivative of the function

Solution By the product rule the derivative is

15. Compute the derivative of

Solution

16. Compute the equation of the line tangent to the graph of at x = 1.

Solution The derivative of the function is . Thus at x = 1 the slope of the tangent line is

The equation of the tangent line is

Solving for y gives

17. The hyperbolic sine function, sinh x, is defined by the formula

The hyperbolic cosine function, cosh x, is defined by the formula

Show that

Solution

Sample Second Midterm Exam with Solutions

Problem One

Compute the derivative with respect to x of

Solution By the quotient rule we have that

Problem Two

Given the implicit equation

Compute the derivative of y with respect to x.

Solution Taking the derivative with respect to x on both sides gives

Solving for the derivative gives

Problem Three

Use a linear approximation to estimate .

Solution The function has derivative . The linear approximation is

Problem Four

Compute the derivative with respect to x of

Solution Grouping the terms as

and then differentiating gives

Problem Five

Compute the derivative with respect to x of

Solution Before differentiating, we can use some of the properties of the log to simplify this expression.

Differentiating gives