Unit 2 and 3 calc derivatives

What is the Mean Value theorem?

if f is continuous, then there is a point where the average rate of change (f(b) - f(a)/b-a) is equal to the instantaneous rate of change. IRC = ARC (plug value into f’(x) to find the x value)

What is the Power Rule?

d/dx x^n = nx^n-1

What is the Product Rule?

d/dx f(x)g(x) = f’(x)g(x) + f(x)g’(x)

What is the Quotient Rule?

d/dx (f(x)/g(x)) = f’(x)g(x) - f(x)g’(x)/g(x)²

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?

sec²x

what is the derivative of cot x?

-csc² x

what is the derivative of sec x?

sec x tan x

what is the derivative of csc x?

-csc x cot x

what is the derivative of ln x?

1/x

what is the derivative of e^x?

e^x

What is the derivative of 5^x? (one we have not learned, works with any constant)

5^x ln x

what is the derivative of sin^-1 x?

1/(sqrt 1-x²)

what is the derivative of cos^-1 x?

-1/(sqrt 1-x²)

what is the derivative of tan^-1 x?

1/1+x²

what is the derivative of cot^-1 x?

-1/1+x²

what is the derivative of sec^-1 x?

1/x(sqrt x²-1)

What is the derivative of csc^-1 x?

-1/x(sqrt x²-1)

What is the chain rule?

derivative of the inside function multiplied by the derivative of the outside function (often use u substitution) f’(g(x)) * g’(x)

What is differentiability?

When you can find a slope at a point, cannot have discontinuities, sharp turns, or vertical asymptotes

How do you find a derivative of a function with both x and y? (aka implicit differentiation)

find dy/dx

What is the equation of a tangent line?

y - y1 = m(x-x1)

What is the equation of a “normal lin/secant line” (perpendicular to tangent line/graph)

y-y1 = -1/m (x-x1) (negative reciprocal of slope)

What are the volumes of a cone, cylinder, and sphere (need to know for general derivatives/integrals)

1/3pir²h pir²h 4/3pir³

How are position, velocity, and acceleration related?

The derivative of position is velocity, and the derivative of velocity is acceleration