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: free variables
• sin(x): sine function
• cos(x): cosine function
• tan(x): tangent function
• sec(x): secant function
• csc(x): cosecant function
• cot(x): cotangent function
• arcsin(x): arcsine function
• arccos(x): arccosine function
• arctan(x): arctangent function
• arcsec(x): arcsecant function
• aliases: asin for arcsin
acos for arccos
atan for arctan,
asec for arcsec
• exp(x): natural exponential function (ex)
• ln(x): natural logarithm function
• log(x): logarithm base 10
• abs(x): absolute value function
• sqrt(x): square root function
These expressions are a subset of the expressions allowed in the other applets -- 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. However, since exponents are handled differently in this applet, the nrt(...) function is also not available (because it is unnecessary -- simply use fractional exponents).

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]
• 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*D[u^(c-1)]
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)*D[y^2] by Constant Multiple Rule
=sin(x)*2*y by Power Rule