# Let me explain Darivative rules...

Gaurav Sharma
19 Points
15 years ago

The Constant Rule
The derivative of a constant function is 0. That is, if c is a real number, then d/dx[c] = 0.

The Sum and Difference Rules
The sum(or difference) of two differentiable functions is differentiable and is the sum(or difference) of their derivatives.

d/dx[f(x) + g(x)] = f'(x) + g'(x)

d/dx[f(x) - g(x)] = f'(x) - g'(x)

The Constant Multiple Rule
If f is a differentiable function and c is a real number, then cf is also differentiable and d/dx[cf(x)] = cf'(x)

The Power Rule
If n is a rational number, then the function f(x) = xn is differentiable and d/dx[xn] = nxn-1

The Product Rule
The product of two differentiable functions, f and g, is itself differentiable. Moreover, the derivative of fg is the first function times the derivative of the second, plus the second function times the derivative of the first.
d/dx[f(x)g(x)] = f(x)g'(x) + g(x)f'(x)

The Quotient Rule
The quotient f/g, of two differentiable functions, f and g, is itself differentiable at all values of x for which g(x) does not = 0. Moreover, the derivative of f/g is given by the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator divided by the square of the denominator.
d/dx[ f(x)/g(x) ] = (g(x)f'(x) - f(x)g'(x)) / [g(x)]2      g(x) does not = 0

The Chain Rule
If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x, then y = f(g(x)) is a differentiable function of x and d/dx[f(g(x))] = f'(g(x))g'(x)

The General Power Rule
If y = [u(x)]n, where u is a differentiable function of x and n is a rational number, then d/dx = [un] = nun-1u'.