IndietroRules of Differentiation and Higher Order Derivatives
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Rules of Differentiation
The Constant Function Rule
The derivative of a constant function is always zero. If f(x) = c, where c is a constant, then:
Formula:
Example: If , then .
The Power Rule
The power rule is used to differentiate functions of the form xn, where n is any real number.
Formula:
Example:
Example:
The Constant Multiple Rule
The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.
Formula:
Example:
The Sum Rule
The derivative of a sum of functions is the sum of their derivatives.
Formula:
The Difference Rule
The derivative of a difference of functions is the difference of their derivatives.
Formula:
Example:
Examples Combining Rules
Example: Differentiate
Higher Order Derivatives
Definition and Computation
Higher order derivatives are obtained by differentiating a function multiple times. The second derivative is the derivative of the first derivative, the third derivative is the derivative of the second derivative, and so on.
Notation: for the second derivative, for the third, and for the nth derivative.
Example: For :
First derivative:
Second derivative:
Third derivative:
Fourth derivative:
Example: Find , , for: a)
b)
Derivatives of Exponential Functions
The Derivative of
The function is differentiable for all real numbers , and its derivative is itself:
Formula:
