Differentiation 2

When do you use chain rule

Composite functions

d/dx f(g(x))

f’(g(x) * g’(x)

Chain rule

F(g(x)) → f’(g(x)) * g’(x)

Product rule

F(x)g(x) → f’(x)g(x) + g’(x)f(x)

F(x)/g(x)

[G(x)f’(x) - f(x)g’(x)] / [g(x)]²

Quotient rule

down d’up - up d’down all over down down

Implicit differentiation

Differentiate term with respect to y then multiply by dy/dx; rearrange for dy/dx =

F(y)

F’(y) * dy/dx

Sin(x)

Cos(x)

Sec(x)

Sec(x)tan(x)

Tan(x)

Sec²(x)

Cos(x)

-sin(x)

Cosec(x)

-cosec(x)cot(x)

Cot(x)

-cosec²(x)

arccos(x)

-1/root(1-x²)

arctan(x)

1/(1+x²)

e^kx

ke^kx

Ln(kx)

1/x

a^kx

a^kx * k[ln(a)]

x = f(t), y = g(t)

g’(t) / f(t)

Implicit differentiation

Dy/dx = dy/dt * dt/dx

When to use implicit differentiation

not in y= form; parametric

When is a function concave

F’’(x) < 0

When is a function convex

F’’(x) > 0

What happens when f’’(x) changes sign

Point of inflection