Chapter 3

Question 1

Derivative of a Constant Function

Answer 1

f1(C) = 0

The derivative of a constant is 0

Question 2

Power Rule

Answer 2

If N is a positive integer, f1(x^n) = nx^n-1

Question 3

Constant Multiple Rule

Answer 3

If C is a constant and f is a differentiable function,

f1 [Cf(x) = Cf1, you don’t differentiate the function

f1(3x^4) = 3(4x^3) 12x^3

Question 4

Sum Rule

Answer 4

if f and g are differentiable

f1 [f(x) + g(x)] = f1f(x) + g1(x)

Question 5

Difference Rule

Answer 5

if f and g are differentiable

f1[f(x) - g(x)] = f1f(x) - g1(gx)

Question 6

Derivative of Natural Log

Answer 6

f1(e^x) = e^x

Question 7

Derivative of a Fraction with X in the Denominator

Answer 7

f(x) = 1/x^2

– f1(x) = x^-2 OR -2x^3 or 2/x^3

Question 8

Product Rule

Answer 8

If and g are differentiable

f1[f(x)g(x)] = f(x)g1 + g(x) f1

Question 9

Quotient Rule

Answer 9

If f and g are differentiable

f1[f(x)/g(x)] = [g(x)f1 - f(x)g1]/g^2

Question 10

Derivatives of Trig Functions: (sinx)

Answer 10

cos(x)

Question 11

Derivatives of Trig Functions: (cosx)

Answer 11

-sin(x)

Question 12

Derivatives of Trig Functions: (tanx)

Answer 12

sec^2x

Question 13

Derivatives of Trig Functions: (cscx)

Answer 13

-csc(x)cot(x)

Question 14

Derivatives of Trig Functions: (secx)

Answer 14

sec(x)tan(x)

Question 15

Derivatives of Trig Functions: (cotx)

Answer 15

-csc^2x

Question 16

Chain Rule

Answer 16

(derivative of outside function, leaving inside function alone) * (derivative of the inside function)

Question 17

Chain Rule Example: (x^2 + 5x - 6)^9

Answer 17

9(x^2 + 5x - 6)^8 * (2x + 5)

Question 18

Chain Rule of Trig Function: y = sin(x^2 - 3x)

Answer 18

derivative of outside function: cos(x^2 - 3x)
derivative of inside function: (2x - 3)
y1 = cos(x^2 - 3x) * (2x - 3)

1. Change the trig sign, keep the inside the same
2. Multiply by the derivative of the inside function

Question 19

Double Chain Rule: (1 + cos2x)^2

Answer 19

• (1 + cos2x)^2
• 2(1+cos2x) * (-sin2sx) [need derivative of second part, keep original second part the same]
• 2(1+cos2x)(-sin2x)(-2)
• (-4)(1+cos2x)(-sin2x)