Chapter 3 Differentiation Rules

Question 1

Derivative of a constant function?

Answer 1

d/dx(c) = 0

Question 2

Power rule?

Answer 2

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

Question 3

Contant Multiple Rule?

Answer 3

d/dx [c f(x)] = c d/dx f(x)

Question 4

Sum Rule?

Answer 4

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

Question 5

Difference Rule?

Answer 5

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

Question 6

Derivative of Natural Exponential Function?

Answer 6

d/dx e^x = e^x

Question 7

Product Rule?

Answer 7

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

Question 8

Quotient Rule?

Answer 8

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

Question 9

Derivative of sin x

Answer 9

cos x

Question 10

Derivative of cos x

Answer 10

-sin x

Question 11

Derivative of tan x

Answer 11

sec^2 x

Question 12

Derivative of csc x

Answer 12

-csc x cot x

Question 13

Derivative of sec x

Answer 13

sec x tan x

Question 14

Derivative of cot x

Answer 14

-csc^2 x

Question 15

Chain Rule

Answer 15

If g is differentiable at x and f is differentiable at g(x), then the composite function F = f g defined by F’(x) = f’(g(x)) * g’(x)

Question 16

Power Rule Combined With Chain Rule

Answer 16

d/dx[g(x)]^n = n[g(x)]^n-1 * g’(x)

Question 17

Derivative of General Exponential Functions

Answer 17

d/dx (b^x) = b^x ln b

Question 18

Implicit Differentiation

Answer 18

We don’t solve an equation for y in terms of x in order to find the derivative of y. Instead, you differentiate both sides of the equation with respect to x and then solve the result dy/dx equation.

Question 19

Derivative of a Logarithmic Function

Answer 19

d/dx (log b x) = 1 / x ln b

Question 20

Steps in logarithmic differentiation

Answer 20

1. Take natural logs of both sides of an equation and use the Laws of Logarithms to expand equation
2. Differentiate implicity with respect to X
3. Solve the resulting equation for y’ and replace y by f(x)

Question 21

d/dx sin^-1 x

Answer 21

1/(1-x^2)^1/2

Question 22

d/dx cos^-1 x

Answer 22

- [1/(1-x^2)^1/2]

Question 23

d/dx tan^-1 x

Answer 23

1/ (1+x^2)

Question 24

d/dx csc^-1 x

Answer 24

- [1/x *(x^2+1)^1/2]

Question 25

d/dx cot^-1 x

Answer 25

-[1 / (1+x^2)]