Math 127
Derivative Shortcuts
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]’ = nxn-1 for ever real number n.
In Leibniz' notation: d(xn)/dx = nxn-1.
4) Exponential Rule: [ax]’ = ax ln(a)
In particular, [ex]’ = ex 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).