Chapter 3 Flashcards
Derivative of a Constant Function
f1(C) = 0
The derivative of a constant is 0
Power Rule
If N is a positive integer, f1(x^n) = nx^n-1
Constant Multiple Rule
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
Sum Rule
if f and g are differentiable
f1 [f(x) + g(x)] = f1f(x) + g1(x)
Difference Rule
if f and g are differentiable
f1[f(x) - g(x)] = f1f(x) - g1(gx)
Derivative of Natural Log
f1(e^x) = e^x
Derivative of a Fraction with X in the Denominator
f(x) = 1/x^2
– f1(x) = x^-2 OR -2x^3 or 2/x^3
Product Rule
If and g are differentiable
f1[f(x)g(x)] = f(x)g1 + g(x) f1
Quotient Rule
If f and g are differentiable
f1[f(x)/g(x)] = [g(x)f1 - f(x)g1]/g^2
Derivatives of Trig Functions: (sinx)
cos(x)
Derivatives of Trig Functions: (cosx)
-sin(x)
Derivatives of Trig Functions: (tanx)
sec^2x
Derivatives of Trig Functions: (cscx)
-csc(x)cot(x)
Derivatives of Trig Functions: (secx)
sec(x)tan(x)
Derivatives of Trig Functions: (cotx)
-csc^2x
Chain Rule
(derivative of outside function, leaving inside function alone) * (derivative of the inside function)
Chain Rule Example: (x^2 + 5x - 6)^9
9(x^2 + 5x - 6)^8 * (2x + 5)
Chain Rule of Trig Function: y = sin(x^2 - 3x)
derivative of outside function: cos(x^2 - 3x)
derivative of inside function: (2x - 3)
y1 = cos(x^2 - 3x) * (2x - 3)
- Change the trig sign, keep the inside the same
- Multiply by the derivative of the inside function
Double Chain Rule: (1 + cos2x)^2
- (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)
