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3.1

The Derivative of a Constant Function

ddx(c) = 0

The derivative of any constant is zero.

The Power Rule

ddx(xn) = nxn-1

The power rule is true for all real numbers n .

The Constant Multiple Rule

ddx[cf (x)] = cddxf (x)

if c is a constant and f is a differentiable function.

When new funtions are formed from old functions by addition, subtraction, or multiplication by a constant, their derivatives can be calculated in terms of derivatives of the old functions. In particular, the Constant Multiple Rule formula says that the derivative of a constant times a function is the constant times the derivative of the function.

The Sum Rule

ddx[f(x) + g(x)] = ddxf(x) + ddxg(x)

if f and g are both differentiable, then the derivative of a sum of functions is the sum of the derivatives.

The Difference Rule

ddx[f(x) - g(x)] = ddxf(x) - ddxg(x)

if f and g are both differentiable, then the derivative of a difference of functions is the difference of the derivatives.

Definition of the Number e

limh0eh-1h=1

The function f(x)=e^x is the one whose tangent line at (0.1) has a slope f'(0) that is exactly 1.

Derivative of the Natural Exponential Function e

ddx(ex) = ex

The function f(x) = e^x has the property that it is its own derivative. The geometrical significance of this fact is that the slope of the tangent line to the curve y=e^x is equal to the y-coordinate of the point.

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