Math 127 Derivative Shortcuts

The derivative shortcuts are of two kinds, general rules for combining functions and

differentiation formulas for familiar functions: There are five of each kind.

General Rules:

1) Constant Multiple Rule: [cf(x)]’ = c[f’(x)].

2) Sum/Difference Rule: [f(x) ± g(x)]’ = f’(x) ± g’(x).

3) Chain Rule: [f(g(x))]’ = f’(g(x)) g’(x) (Newton's notation). A bit easier to understand is

Leibniz' alternate notation: If y = f(z) and z = g(t), then dy/dt = (dy/dz) (dz/dt).

4) Product Rule: [f(x) g(x)]’ = f’(x) g(x) + f(x) g’(x). In Leibniz' notation this becomes:

If y = uv where both u and v are functions of x. Then d(uv)/dx = (du/dx)v + u(dv/dx).

5) Quotient Rule: [f(x)/g(x)]’ = [f’(x)g(x) - f(x)g’(x)]/(g(x))² In Leibniz' notation this becomes:

d(u/v)/dx = [(du/dx)v - u(dv/dx)]/v² where u and v are functions of x.

Differentiation Formulas:

1) Derivative of a constant: [c]’ = 0 or dc/dx = 0 in Leibniz form.

2) Derivative of a linear function: [mx + b]’ = m = slope of the line y = mx + b.

3) Power Rule: [xⁿ]’ = nx^n-1 for ever real number n. In Leibniz' notation: d(xⁿ)/dx = nx^n-1.

4) Exponential Rule: [a^x]’ = a^x ln(a) In particular, [e^x]’ = e^x whenever a = e = 2.71828…

5) Logarithm Rule: [ln(x)]’ = 1/x or in Leibniz' notation d(ln(x))/dx = 1/x.

Finally, for the sake of completeness we include a pair of formulas which are not in our syllabus:

6) Trigonometric Rules: [sin(x)]’ = cos(x) and [cos(x)]’ = -sin(x).