

Derivatives
The derivative of a function is the instantaneous rate of change of the function with respect to one or more
variable parameters. In general, if \(y=f(x)\) and the variable parameter \(x\) is given a rate of change \(Δx\),
then:
While calculating the derivative of a function using the formula above will always work, elementary rules exist for quick differentiation of certain functions. In general, where \(y=x^n\): Since the derivative of a function is a measure of the rate of change of the function, the derivative of a constant is zero. Also, the derivative of a variable with respect to itself is always 1. Any constant multiplier remains a multiplier in the derivative, while added constants disappear. If there are added terms, each term is differentiated separately. For example, if \(y=4x^2\frac{x}{3}+2\), the derivative is: The derivative of a function may be written in various forms. For example, \(\frac{dy}{dx}\), \(u^′(x)\), and \(Dx\), are all representative forms of the derivative. Which one to use is entirely up to style and clarity of the moment. Derivative of a Constant
If \(f\) is a function such that \(f(x)=c\), then the derivative is zero. In symbolic terms:
Derivative of \(x^n\) (Power Rule)
If \(f\) is a function such that \(f(x)=x^n\), then the derivative is the expontent times \(x\) to the power \(n1\). Note that
this is true in all cases, including when \(n=0\). The power rule is generally written:
Derivative of a Sum (Sum Rule)
If \(u\) and \(v\) are differentiable at \(x\) and \(f\) is a function such that \(f(x)=u(x)+v(x)\), then the derivative is
the sum of the derivative of \(u(x)\) and the derivative of \(v(x)\). In general:
Derivative of a Product (Product Rule)
If \(u\) and \(v\) are differentiable at \(x\) and \(f\) is a function such that \(f(x)=u(x)\cdot v(x)\), then the derivative is the first factor
times the derivative of the second, plus the second factor times the derivative of the first. In general:
Derivative of a Constant Times a Function
If \(u\) is differentiable at \(x\) and \(f\) is a function such that \(f(x)=c\cdot u(x)\), then the derivative is the constant times the
derivative of the function:
Derivative of a Quotient (Quotient Rule)
If \(u\) and \(v\) are differentiable at \(x\) and \(f\) is a function such that \(f(x)=\frac{u(x)}{v(x)}\), then the derivative is the denominator
times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator.
Chain Rule
If \(f\) and \(g\) are functions such that \(g\) is differntiable at \(x\) and \(f\) is differntiable at \(u=g(x)\) and \(F\) is a function such that
\(F(x)=f(g(x))\), then the derivative of \(F(x)\) is the derivative \(f(u)\) times the derivative of \(g(x)\):
Power Rule
If \(u\) is a function that is differentiable at \(x\) and \(f\) is a function such that \(f(x) = [u(x)]^n\), where \(n\) is a rational number,
then the derivative is the exponent times \(u(x)\) to the power \(n1\), times the derivative of \(u(x)\):
TablesTable of Derivatives 

Calculus: Introduction  Calculus: Integrals 