### 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: 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 (xy)
• ( ... ): 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 (ex)
• 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