Derivatives Review (Part 1)

General Knowledge (Vocabulary, Symbols, etc...)

100

The process of making something easier to do or understand.

What is to simplify?

200

A line that just touches a curve at a point, matching the curve's slope there.

What is a tangent line?

300

This is the rate of change of a function with respect to a variable.

What is a derivative?

400

This is what I would get if I took the derivative of 

f⁽⁵⁾(x).

What is the 6th derivative?

500

The notation for this can be written like so,

(d⁴y)/(dx⁴).

What is the 4th derivative?

You Better Know it! (Formulas)

100

This is the formula for the Product Rule.

What is,

d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)?

200

This is the formula for the Quotient Rule.

What is,

d/dx[f(x)/g(x)] = [f'(x)g(x) - f(x)g'(x)]/[g(x)]²?

300

That is the formula for Chain Rule.

What is,

d/dx[(f(x))ⁿ] = n[f(x)]^n-1 * [f(x)]'?

400

These are the derivatives of cot(x) and csc(x).

What is,

d/dx[cot(x)] = -csc²(x)

and

d/dx[csc(x)] = -csc(x)cot(x)?

500

WORTH DOUBLE POINTS!!!

This is the formula for a^x.

What is,

d/dx(a^x) = a^x ln(a) (x)'?

Back to the Beginning (Section 3.3/3.4)

100

This is the first derivative to the following problem:

y = 3x³ - 4x² + 1000.

What is,

y' = 9x² - 8x?

200

WORTH DOUBLE POINTS!!

This is the answer to the second derivative (simplified) of the following problem:

y = xe^3x.

What is,

y' = (3x + 1)e^3x 

y'' = (9x + 6)e^3x?

300

This is the velocity of the following problem:

s(t) = 179 - 16t².

What is,

v(t) = s'(t) = -32t?

400

This is the answer to the following question:

If a ball is moving at s(t) = 179 - 16t², how long would it take for the ball to hit the ground?

What is,

t = square root(179/16)?

500

WORTH TRIPLE POINTS

This is the first derivative (simplified) to the following problem:

y = (x³ + e^x)/(5x⁶).

What is,

y' = [e^x(x - 6) - 3x³]/[5x⁷]?

Trig and Chain Rule and Factoring! Oh my! (Section 3.5/3.6)

100

This is the answer to the third derivative of the following problem:

y = sin(2x) + cos(2x).

What is,

y' = 2cos(2x) - 2sin(2x)

y'' = -4sin(2x) - 4cos(2x)

y''' = -8cos(2x) + 8sin(2x)?

200

This is the answer to the first derivative of the following problem:

y = sec(tan(x)).

What is,

y' = sec²(x)sec(tan(x))tan(tan(x))?

300

This is the answer to the first derivative of the following problem:

y = (tan(e^x))³

What is,

y' = 3e^xsec²(e^x)tan²(e^x)?

400

WORTH DOUBLE POINTS

This is the answer to the first derivative (simplified) to the following problem:

y = (4x⁵ + 3x²)^-3

What is,

y = [-6x(10x³ + 3)]/[(4x⁵ + 3x²)⁴]

500

ANTY UP (EVERYONE)

This is the answer to the first derivative (simplified) of the following problem:

y = x³e^csc(3x).

What is,

y = -3x²e^csc(3x)(xcot(3x)csc(3x) - 1)?