Test 2 Review

Basic Derivatives

100

Find f'(t).

f(t) = -3t²+2t-4

What is f'(t) = -6t+2?

200

Find f'(x) given 

f(x) = x^1/2-3ln(x)

What is 

f'(x) = 1/2x^-1/2-3/x?

300

Find g'(x) 

given g(x) = -2e^x - 4x² + 8x - 9 

What is g'(x) = -2e^x - 8x + 8?

400

Find f'(x) given 

f(x) = x^5/2 - x^4/3 + x^1/4

What is 

f'(x) = (5/2)x^3/2 - (4/3)x^1/3 + (1/4x^3/4)

500

What is the d/dx[(pi)⁵]?

What is 0?

Product Rule

100

Find the derivative of 

(3-x²)(x+4)

What is -3x²-8x+3?

200

Find the derivative of

x²*sin(x)

What is x²cos(x) + 2x*sin(x)?

300

Find the derivative

e^x(x²+4x)

What is e^x(2x+4) + e^x(x²+4x)?

400

Find the derivative of

x²ln(x)

What is 2xln(x) + x?

500

What is the derivative of 

sin(x)sec(x)?

Write your answer in simplest form.

What is sec²(x)?

Quotient Rule

100

What is the derivative of 

(3x-4)/(2x²-1)?

Simplify your answer. 

What is -6x^2?

200

What is the derivative of 

tan(x)/(x² + 5)?

Simplify your answer.

What is

(x²sec²(x) + 5sec²(x) - 2xtan(x))

----------------------------------------

                     (x²+5)²

300

What is the derivative of 

e^x/x²?

Simplify your answer.

What is e^x(x-2)?

400

What is the derivative of 

ln(x)/(3x²)?

Simplify your answer.

What is 

(1-2ln(x))/(3x³)

500

What is the derivative of 

sec(x)/ln(x)?

Express your answer without fractions in the numerator. 

What is 

xsec(x)tan(x)ln(x) - sec(x)

------------------------------

               x*ln²(x)

Trig Derivatives

100

What is the derivative of cos(x)?

What is -sin(x)?

200

What is the derivative of sec(x)?

What is sec(x)tan(x)?

300

What is the derivative of csc(x)?

What is -csc(x)cot(x)?

400

What is the derivative of tan(x)?

What is sec²(x)?

500

What is the derivative of cot(x)?

What is -csc²(x)?

Applications

100

Derivatives are used to find this in the equation of a tangent line.

What is the slope?

200

Equations that are normal to a curve have what slope relationship? 

What is the opposite reciprocal? 

300

What are the three conditions where a function is continuous but not differentiable? 

What is corner, cusp, and vertical tangent?

400

Complete the statement:

_______________ does not imply differentiability.

What is "continuity"?

500

What are the conditions to prove a function is differentiable at x = a?

What is f(x) is continues at x = a and the limit as x approaches a from the left of f'(x) equals the limit as x approaches a from the right of f'(x).

Note: The slopes must match from both sides (i.e. both must be positive).