AP Calc BC Unit 2 Concepts

What is a derivative?

Instantaneous rate of change

Derivative at a specific point c

f’(x) = lim_h—>0 f(x)-f(c ) / x-c

General derivative formula

f’(x) = lim_h—>0 f(c+h)-f(c ) / h
f’(x) = lim _∆x—>0 f(x+∆x)-f(x) / ∆x

Theorem: Differentiability Implies Continuity

If f is differentiable at x=c, meaning if there is a derivative at x=c, then f is continuous at x=c

Theorem: Tangent Line Equation

If m_tan exists, then the tangent line equation to the graph of f is y-f(c)=f’(c)(x-c).

The Constant Rule

d/dx[c] = …

The derivative of the constant function is 0, meaning if c is a real #, d/dx[c] = 0

The Power Rule

d/dx[xⁿ] = …

If n is a rational #, then f(x)=xⁿ is differentiable and d/dx[xⁿ] = nx^n-1. For f to be differentiable at x=0, n must be a # such that x^n-1 is defined on an open interval containing 0.

d/dx[x] = …

1
…because d/dx[x¹] = (1)(x^1-1) = x⁰ = 1

The Sum & Difference Rules'
d/dx[f(x)+g(x)] = …
d/dx[f(x) - g(x)] = …

d/dx[f(x)+g(x)] = f’(x)+g’(x)

d/dx[f(x) - g(x)] = f’(x) - g’(x)

d/dx[sinx] = …

d/dx[sinx] = cosx

d/dx[cosx] = …

d/dx[cosx] = -sinx

d/dx[e^x] = …

d/dx[e^x] = e^x

d/dx[lnx] = …

d/dx[lnx] = 1/x

d/dx[log_b x] = …

d/dx[log_b x] = 1/(ln bx)

The Product Rule

d/dx[f(x)g(x)] = …

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

The Quotient Rule

d/dx[f(x)/g(x)] = …

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

d/dx[tanx] = …

d/dx[tanx] = sec²x

d/dx[cotx] = …

d/dx[tanx] = -csc²x

d/dx[secx] = …

d/dx[secx] = secxtanx

d/dx[cscx] = …

d/dx[cscx] = -cscxcotx

sin²θ + cos²θ =

1

1+tan²θ =

sec²θ

1 + cot²θ =

csc²θ