Basic Differentiation Formulas



DERIVATIVE OF A CONSTANT FUNCTION:




EXAMPLES:


THE POWER RULE: If is a positive integer, then




EXAMPLES: Differentiate

(a)


(b)


(c)



THE POWER RULE (GENERAL VERSION): If is any real number, then



EXAMPLES: Differentiate

(a)


(b)


(c)


(d)


(e)


(f)


(g)



THE CONSTANT MULTIPLE RULE: If is a constant and is a differentiable function, then




EXAMPLE: Differentiate .


EXAMPLE: Find equations of the tangent line and normal line to the curve at the point .



THE SUM/DIFFERENCE RULE: If and are both differentiable functions, then



or




EXAMPLE: Differentiate .


EXAMPLE: Differentiate .


REMARK: We can combine two previous rules in one. If are constants and are both differentiable functions, then



or



EXAMPLE: Differentiate .



THE DERIVATIVE OF THE SINE AND COSINE FUNCTIONS: We have



and




EXAMPLE: Differentiate .


EXAMPLE: Find if .