AP Calculus: Derivatives and Formulas

What is the derivative of e^ax ?

a(e^ax)

What is the derivative of 1/x ?

-1/x²

What is the derivative of (x)^1/2?

1/2(x)^-1/2

What is the derivative of an absolute value function?

#, x >0 and # x<0

(but undefined when the function equals a certain value)

f(x) = g(x) + or - h(x)

f’(x) = g’(x) + or - h’(x)

If a function is differentiable at a point/on an interval, then it is ___ at that point/on that interval?

Continuous

T&F: If a function is continuous, then it is differentiable.

False (differentiability = continuity, but continuity does not equal differentiability)

What things would make a function not differentiable? (types of points? tangents?)

All corners & curved points (cusps), vertical tangents, end points, and discontinuities.

When an object is moving up, what sign is its velocity? moving down? 

positive when moving up and negative when moving down. 

When an object is accelerating up, what sign is its acceleration? accelerating down?

Sign is positive when accelerating up and negative when accelerating down. 

How do you know if an object is speeding up or slowing down (velocity (V) and acceleration (A) explanation)?

If (V x A) = a number > 0, therefore object is speeding up.

If (V x A) = a number < 0, therefore object is slowing down.

T&F: You use velocity to find whether or not an object is speeding up and acceleration to identify the object’s direction of motion.

False

(velocity is used to identify an objects direction of motion and acceleration is used to find out if an object is speeding up or slowing down.)

When does an object have a velocity of 0?

When it is at rest (not moving, including peak/turning point of motion).

What is the derivative of sin(x)?

cos(x)

What is the derivative of cos(x)?

-sin(x)

What is the derivative of tan(x)?

sec²(x)

What is the derivative of csc(x)?

-csc(x)cot(x)

What is the derivative of sec(x)

sec(x)tan(x)

What is the derivative of cot(x)?

-csc²(x)

What is the derivative of a^f(x) ?

(ln(a))(a^f(x))(f’(x))

What does linearization refer to?

equation of the tangent line

What is the derivative of an inverse function?

g’(x) = 1/f’(g(x))

When f(x) & g(x) are inverses of eachother

What is the derivative of y = log_a(x)

y = 1/(x)(ln(a))

What is the derivative of y = ln(x)

y = 1/x

What is the derivative of sin^-1(x) ?

1/(1-x²)^1/2

What is the derivative of cos^-1(x) ?

-1/(1-x²)^1/2

What is the derivative of tan^-1(x) ?

1/x²+1