NOTE: Throughout this document, a and c represent constants; x is the
variable in all cases (it's not a bad idea to change the names of your
variables to help yourself learn, by the way). There are four columns
below. Reading across, the columns give you a basic function, its
derivative, a "chain rule version" of the basic function, and the
derivative of the "chain rule version". Examples are included at
the end of most sections.
The deriv of is Chain rule version yields
============ ====== ================== ======
c 0
x 1
c*x c c*f(x) c*f'(x)
x^n (n not 0) nx^(n-1) [f(x)]^n n([f(x)]^(n-1)) * f'(x)
two examples:
x^(-3) -3*[x^(-4)] [5x]^9 9[(5x)^8] * 5
Note: The stuff "inside" being raised to the power NEVER CHANGES on the
"inside". It's just being raised to one less power (i.e. In the
example above, 5x stays 5x, it just ends up being raised to the
8th power instead of 9th). Changing that function on the inside
is a common mistake.
-----------------------------------------------------------------------------
The deriv of is Chain rule version yields
============ ====== ================== ======
Sin[x] Cos[x] Sin[f(x)] (Cos[f(x)]) * f'(x)
one example: Sin[x^4] (Cos[x^4])*(4*x^3)
Cos[x] -Sin[x] Cos[f(x)] -(Sin[f(x)]) * f'(x)
one example: Cos[x^7] -(Sin[x^7])*(7*x^6)
Note: The stuff "inside" the trig function NEVER CHANGES on the "inside".
Changing the function inside is a common mistake.
-----------------------------------------------------------------------------
The deriv of is Chain rule version yields
============ ====== ================== ======
Log[x] 1/x Log[f(x)] [1/f(x)] * f'(x) which
(this is "log base e", remember) is simplified to f'(x)/f(x)
one example: Log[Cos(x)] -Sin[x]/Cos[x]
-----------------------------------------------------------------------------
The deriv of is Chain rule version yields
============ ====== ================== ======
E^x E^x E^f(x) [E^f(x)] * f'(x)
Note: The stuff in the "power" of the original exponential function
NEVER CHANGES in the power. Changing that function is a common mistake.
one example: E^Sin[x] [E^Sin[x]] * Cos[x]
-----------------------------------------------------------------------------
SOME GENERAL RULES:
The deriv of is
============ ======
c*f(x) c*f'(x) (Seen above)
one example:
10*E^(2x) 10*([E^(2x)]*2)
(We are asking for the deriv of 10*E^f(x) here, where f(x) = 2x)
-----------------------------------------------------------------------------
The deriv of is
============ ======
f(x) + g(x) f'(x) + g'(x) (Sum rule)
f(x) - g(x) f'(x) - g'(x) (Difference rule)
one long example:
x^2 + 5 + Cos(4x) - Log[3x+1] 2x + 0 -4*Sin(4x) - 3/(3x+1)
-----------------------------------------------------------------------------
The deriv of is
============ ======
f(x)*g(x) f'(x)*g(x) + f(x)*g'(x) (Product rule)
one example:
(x^2 + 4x)*(Sin[x-3]) (2x + 4)*(Sin[x-3]) + (x^2 + 4x)*(Cos[x-3])
Note: The deriv of Sin[x-3] is really a deriv of Sin[f(x)] where f(x) = x-3,
but the deriv of x-3 is 1, so it doesn't "appear" in the answer.
-----------------------------------------------------------------------------
The deriv of is
============ ======
[f(x)]^n n([f(x)]^(n-1)) * f'(x) (Special version of Chain Rule.
Seen above already.)
another example of this:
[3x - Cos(2x)]^(4.5) (4.5)*[[3x - Cos(2x)]^(3.5)]*[3 + 2*Sin(2x)]
Note: The stuff "inside" being raised to the power NEVER CHANGES on the
"inside". It's just being raised to one less power. Changing that
function on the inside is a common mistake.
-----------------------------------------------------------------------------
The deriv of is
============ ======
f(x)/g(x) EITHER A) rewrite f(x)/g(x) as f(x) * [g(x)^(-1)] and use
the previous two rules. This yields:
[f'(x)*[g(x)^(-1)]] + [f(x)*[-1*g(x)^(-2)]*g'(x)]
OR B) [g(x)*f'(x) - f(x)*g'(x)]/[g(x)^2]
Both answers are equivalent here. (Get a common denominator in the answer to A,
remembering that (thing)^(-1) = 1/(thing).) The rule in answer B is called the
Quotient rule.
One example, shown both ways:
Sin[x]/Log[x] A) rewrite: Sin[x]/Log[x] = Sin[x] * [(Log[x])^(-1)]
Use product rule, yielding:
Cos[x]*(Log[x])^(-1) + Sin[x]*(-1)*[(Log[x])^(-2)]*(1/x)
B) [(Log[x])*Cos[x] - Sin[x]*(1/x)]/(Log[x])^2
