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3.1

The Derivative of a Constant Function

$\frac{d}{d x} (c) = 0$

The derivative of any constant is zero.

The Power Rule

$\frac{d}{d x} (x^{n}) = n x^{n - 1}$

The power rule is true for all real numbers n .

The Constant Multiple Rule

$\frac{d}{d x} [c f (x)] = c \frac{d}{d x} f (x)$

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

$\frac{d}{d x} [f (x) + g (x)] = \frac{d}{d x} f (x) + \frac{d}{d x} g (x)$

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

The Difference Rule

$\frac{d}{d x} [f (x) - g (x)] = \frac{d}{d x} f (x) - \frac{d}{d x} g (x)$

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

$\lim_{h \to 0} \frac{e^{h} - 1}{h} = 1$

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

$\frac{d}{d x} (e^{x}) = e^{x}$

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.

3.1

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