Rules of Differentiation

Constant Rule

The Constant Rule states that the derivative of any constant function is zero. This is because a constant function does not change, so its rate of change is zero.

• Formula:

• Example:

Power Rule

The Power Rule is used to differentiate functions of the form , where is a real number. The rule states that the derivative is times raised to the power .

• Formula:

• Example:

Constant Multiple Rule

The Constant Multiple Rule states that the derivative of a constant times a function is the constant times the derivative of the function.

• Formula:

• Example:

• Example:

Sum and Difference Rule

The Sum/Difference Rule states that the derivative of a sum or difference of functions is the sum or difference of their derivatives.

• Formula:

Derivative of the Exponential Function

The derivative of the exponential function is itself.

• Formula:

Examples of Differentiation

Applying the above rules to various functions:

• Example 1:

• Example 2:

• Example 3:

• Example 4:

• Example 5:

Application: Slope of Tangent Line

To find the point on a curve where the slope of the tangent line equals a given value, set the derivative equal to that value and solve for .

• Example: For , find where the slope is 16. Set Point: (2, 8)

Application: Horizontal Tangent Lines

Horizontal tangent lines occur where the derivative is zero.

• Example: For Set Solutions:

Application: Equation of Tangent Line

To find the equation of the tangent line at a point, use the point-slope form with the derivative evaluated at the given value.

• Example: For at (1, 2): Equation:

Higher Order Derivatives

The second derivative measures the rate of change of the first derivative, often related to concavity and acceleration. The nth derivative is the result of differentiating a function times.

• Second Derivative Notation: or

• nth Derivative Notation: or

Example: First and Second Derivatives

Find the first and second derivatives of a polynomial function.

• Example: