Unit 2 - Differentiation: Definition and Definitive Properties

Question 1

Average Rate of Change

Answer 1

[f(x+h)-f(x)] / [h]

This gives the gradient of the curve

Question 2

Derivative notation

Answer 2

lim x–>h [f(x+h) - f(x)] / h

Question 3

Derivative of a constant

Answer 3

0 [Slope is 0]

Question 4

Power rule (derivative)

Answer 4

x^n = nx^(n-1)

Question 5

Derivative of e^x

Answer 5

e^x

Question 6

Derivative of ln(x)

Answer 6

1/x

Question 7

derivative of:
1. sin(x)
2. cos(x)
3. tan(x)
4. cot(x)
5. sec(x)
6. cosec(x)

Answer 7

1. cos(x)
2. -sin(x)
3. sec^2(x)
4. -cosec^2(x)
5. sec(x)tan(x)
6. -cosec(x)cot(x)

Question 8

Derivative of:
1. arcsin()
2. arccos()
3. arctan()
4. arcsec()
5. arccosec()
6. arccot()

Answer 8

1. 1 / root (1-x^2)
2. -1 / root (1-x^2)
3. 1 / (1 + x^2)
4. 1 / |x|root (x^2 - 1)
5. -1 / |x|root (x^2 - 1)
6. -1 / (1 + x^2)

Question 9

Odd even function

Answer 9

Odd: f(-x) = -f(x)
Even: f(-x) = f(x)

Question 10

Derivative of mod

Answer 10

-1 when x<0
1 when x>0

Question 11

When are derivatives reciprocal

Answer 11

At points where the two functions are inverse of each other

Question 12

Rolle’s Theorem

Answer 12

If a real-valued function f is continuous on a proper closed interval [a, b], differentiable on the open interval (a, b), and f (a) = f (b), then there exists at least one c in the open interval (a, b) such that
f’(c)=0.

Question 13

When do derivatives not exist?

Answer 13

• Functions which are not continuous
• Functions which have a slope of infinity (vertical slope)

Question 14

Derivative of b^x

Answer 14

derivative = b^x (B derivative function (0))