Expressions and Derivative Rules for the Derivative Calculator
Expressions
The expressions entered into the text field
in the Derivative Calculator
may include any combination
of the following:
- x + y: addition
- x - y: subtraction
- x * y: multiplication
- In many situations, the asterisk "*" is not necessary in
multiplication expressions -- for example, the expression
"2pi" is correct
- x / y: division
- x ^ y: exponentiation (x^{y})
- ( ... ): grouping
- decimal numbers, e.g. 5.317
- pi: constant (3.1416...)
- e: constant (2.7183...)
- x, y, z,
u, v, r,
s, t: free variables
- sin(x): sine function
- cos(x): cosine function
- tan(x): tangent function
- arcsin(x): arcsine function
- arccos(x): arccosine function
- arctan(x): arctangent function
- aliases: asin for arcsin
acos for arccos
atan for arctan
- exp(x): natural exponential function
(e^{x})
- ln(x): natural logarithm function
- abs(x): absolute value function
- sqrt(x): square root function
These expressions are a subset of
the expressions allowed in other Mathlets
-- in particular, expressions in the Derivative Calculator do not include
the functions
int(...),
max(...), or
min(...),
factorial expressions n!, or
conditional expressions
x?y:z.
These restrictions are mostly intended to eliminate potential problems
with derivatives at endpoints of intervals or jump discontinuities.
Derivative Rules
In these rules, the derivative operator is
D[...],
u and v are expressions involving the
free variable x, and c is a constant expression.
- Constant Rule:
D[c] = 0
- Derivative of x:
D[x] = 1
If u = x in below rules, resulting
D[x] factor is usually dropped
- Constant Multiple Rule:
D[c*u]
= c*D[u]
special case:
D[-u] = -D[u] (constant multiple of -1)
- Sum Rule:
D[u+v]
= D[u]+D[v]
special case:
D[u+c]
= D[c+u]
= D[u]
- Product Rule:
D[u*v]
= u*D[v]
+v*D[u]
- Quotient Rule:
D[u/v]
= (v*D[u]
- u*D[v])
/(v^2)
special case:
D[c/u]
= - c*D[u]/u^2
(as application of Power Rule)
- Power Rule:
D[u^c]
= c*u^(c-1)*D[u]
special case:
D[u^2]
= 2*u*D[u]
- Logarithmic Differentiation:
D[u^v]
= (u^v)*(
v*D[u]/u+
ln(u)*D[v])
- Chain Rule:
D[f (u)]
= f '(u)*D[u],
where f (x) is a function and f '(x)
is its derivative
e.g. Sine Rule with Chain Rule:
D[sin(u)] =
cos(u)*D[u]
Rules below all have Chain Rule forms, but written here without Chain Rule
- Sine Rule:
D[sin(x)] = cos(x)
- Cosine Rule:
D[cos(x)] = - sin(x)
- Tangent Rule:
D[tan(x)] = (sec(x))^2
- Secant Rule:
D[sec(x)] = sec(x)*tan(x)
- Cosecant Rule:
D[csc(x)] = - csc(x)*cot(x)
- Cotangent Rule:
D[cot(x)] = - csc(x)^2
- Arcsine Rule:
D[arcsin(x)] = 1/sqrt(1-x^2)
- Arccosine Rule:
D[arccos(x)] = -1/sqrt(1-x^2)
- Arctangent Rule:
D[arctan(x)] = 1/(1+x^2)
- Arcsecant Rule:
D[arcsec(x)] = 1/(x*sqrt(x^2-1))
- Exponential Rule:
D[exp(x)] = exp(x)
- Logarithm Rule:
D[ln(x)] = 1/x
- Logarithm Rule -- base 10:
D[log(x)] = 1/(x*ln(10))
- Absolute Value Rule:
D[abs(x)] = abs(x)/x
- Square Root Rule: (as application of Power Rule)
D[sqrt(x)] = 1/(2*sqrt(x))
Partial Derivatives
The standard derivative operator D[...]
represents a derivative with respect to the free variable x.
This derivative operator can also be entered as
Dx[...]. If other free variables appear
in the expression, they are treated as constant (with respect to
the free variable x), so that the derivative computed is
actually a partial derivative with respect to x.
Partial derivatives with respect to other available free variables can
be computed by using the appropriate derivative operator:
Dy[...] represents a partial derivative
with respect to the free variable y,
Dz[...]
with respect to z, etc.
Available free variables are: x, y,
z, u, v, r,
s, and t.
Examples
The following examples show partial derivatives of the same expression
with respect to different variables.
- Dx[y^2*sin(x)]=y^2*Dx[sin(x)]
by Constant Multiple Rule
=y^2*cos(x) by Sine Rule
- Dy[y^2*sin(x)]=sin(x)*Dy[y^2]
by Constant Multiple Rule
=sin(x)*2*y by Power Rule