AP Calculus AB - Final Exam Study Guide
Definition of a Derivative - Answer>>f'(x) = lim (h goes to zero) [(f(x+h) - f(x))/h]
Alternate Form of a Derivative - Answer>>f'(a) = lim (x goes to a) [(f(x) - f(a))/(x-a)]
Constant Rule (Definition) - Answer>>If f(x) = c, where c is any real number, then f'(x) = 0.
Power Rule (Definition) - Answer>>If f(x) = xⁿ, then f'(x) = nxⁿ ⁻¹, where n∈R.
Constant Multiple Rule (Definition) - Answer>>If y = cf(x), where c
is any real number, then dy/dx = cf'(x).
The Sum and Difference Rules (Definition) - Answer>>The derivative of a sum or difference is the sum or difference of the derivatives.
Sum Rule (Notation) - Answer>>d/dx[f(x) + g(x)] = f'(x) + g'(x)
Difference Rule (Notation) - Answer>>d/dx[f(x) - g(x)] = f'(x) - g'(x)
Derivative of eⁿ (Definition) - Answer>>If f(x) = eⁿ, then f'(x) = eⁿ
Derivative of ln(x) (Definition) - Answer>>If f(x) = ln(x), then f'(x) = 1/x
Product Rule (Notation) - Answer>>d/dx[f(x) × g(x)] = gf' + fg' Quotient Rule (Notation) - Answer>>d/dx[f(x)/(gx)] = (gf' - fg')/g²
9th Rule: Cosine - Answer>>If f(x) = cos(x), then f'(x) = -sin(x)
10th Rule: Sine - Answer>>If f(x) = sinx(x), then f'(x) = cosx
11th Rule: Tangent - Answer>>If f(x) = tan(x), then f'(x) = sec²(x)
12th Rule: Cotangent - Answer>>If f(x) = cot(x), then f'(x) = -
csc²(x)
13th Rule: Secant - Answer>>If f(x) = sec(x), then f'(x) = tan(x)sec(x)
14th Rule: Cosecant - Answer>>If f(x) = csc(x), then f'(x) = -
cot(x)csc(x)
Chain Rule - Answer>>dy/dx = [f'(g(x))][g'(x)]
Exponential Function Rule - Answer>>If f(x) = aⁿ, then f'(x) = aⁿ+ln(a)
Logarithmic Function Rule - Answer>>If f(x) = log(a), then f'(x) = 1/xln(a)
18th Rule: Arcsin - Answer>>d/dx sin ⁻¹(x) = (1/(√1-u²))(du/dx), |u|
<1
19th Rule: Arccos - Answer>>d/dx cos ⁻¹(x) = -(1/(√1-u²))(du/dx), |
u|<1
20th Rule: Arctan - Answer>>d/dx tan ⁻¹(x) = (1/(1+u²))(du/dx), u is ARN
Definition of a Derivative - Answer>>f'(x) = lim (h goes to zero) [(f(x+h) - f(x))/h]
Alternate Form of a Derivative - Answer>>f'(a) = lim (x goes to a) [(f(x) - f(a))/(x-a)]
Constant Rule (Definition) - Answer>>If f(x) = c, where c is any real number, then f'(x) = 0.
Power Rule (Definition) - Answer>>If f(x) = xⁿ, then f'(x) = nxⁿ ⁻¹, where n∈R.
Constant Multiple Rule (Definition) - Answer>>If y = cf(x), where c
is any real number, then dy/dx = cf'(x).
The Sum and Difference Rules (Definition) - Answer>>The derivative of a sum or difference is the sum or difference of the derivatives.
Sum Rule (Notation) - Answer>>d/dx[f(x) + g(x)] = f'(x) + g'(x)
Difference Rule (Notation) - Answer>>d/dx[f(x) - g(x)] = f'(x) - g'(x)
Derivative of eⁿ (Definition) - Answer>>If f(x) = eⁿ, then f'(x) = eⁿ
Derivative of ln(x) (Definition) - Answer>>If f(x) = ln(x), then f'(x) = 1/x
Product Rule (Notation) - Answer>>d/dx[f(x) × g(x)] = gf' + fg' Quotient Rule (Notation) - Answer>>d/dx[f(x)/(gx)] = (gf' - fg')/g²
9th Rule: Cosine - Answer>>If f(x) = cos(x), then f'(x) = -sin(x)
10th Rule: Sine - Answer>>If f(x) = sinx(x), then f'(x) = cosx
11th Rule: Tangent - Answer>>If f(x) = tan(x), then f'(x) = sec²(x)
12th Rule: Cotangent - Answer>>If f(x) = cot(x), then f'(x) = -
csc²(x)
13th Rule: Secant - Answer>>If f(x) = sec(x), then f'(x) = tan(x)sec(x)
14th Rule: Cosecant - Answer>>If f(x) = csc(x), then f'(x) = -
cot(x)csc(x)
Chain Rule - Answer>>dy/dx = [f'(g(x))][g'(x)]
Exponential Function Rule - Answer>>If f(x) = aⁿ, then f'(x) = aⁿ+ln(a)
Logarithmic Function Rule - Answer>>If f(x) = log(a), then f'(x) = 1/xln(a)
18th Rule: Arcsin - Answer>>d/dx sin ⁻¹(x) = (1/(√1-u²))(du/dx), |u|
<1
19th Rule: Arccos - Answer>>d/dx cos ⁻¹(x) = -(1/(√1-u²))(du/dx), |
u|<1
20th Rule: Arctan - Answer>>d/dx tan ⁻¹(x) = (1/(1+u²))(du/dx), u is ARN
