Summary of Derivative Rules. Spring 2012. 1 General Derivative Rules. 1. Constant Rule d dx. [c]=0. 2. Constant Multiple Rule d dx. [cf (x)] = cf (x). 3. Sum Rule d.
Summary of Derivative Rules
1
Spring 2012
General Derivative Rules
1. Constant Rule
d [c ] = 0 dx
2. Constant Multiple Rule
d [cf (x )] = cf 0 (x ) dx
3. Sum Rule
d [f (x ) + g (x )] = f 0 (x ) + g 0 (x ) dx
4. Difference Rule
d [f (x ) − g (x )] = f 0 (x ) − g 0 (x ) dx
5. Product Rule 6. Quotient Rule
7. Chain Rule
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d [f (x )g (x )] = f 0 (x )g (x ) + f (x )g 0 (x ) dx d f (x ) g (x )f 0 (x ) − f (x )g 0 (x ) = 2 dx g (x ) [g (x )] d [f (g (x ))] = f 0 (g (x ))g 0 (x ) dx
Derivative Rules for Particular Functions Basic Rule
Chain Rule Form
1. Powers
d n [x ] = nx n−1 dx
d [(f (x ))n ] = n(f (x ))n−1 f 0 (x ) dx
2. Sine
d [sin x ] = cos x dx
d [sin (f (x ))] = cos (f (x ))f 0 (x ) dx
3. Cosine
d [cos x ] = − sin x dx
d [cos (f (x ))] = − sin (f (x ))f 0 (x ) dx
4. Tangent
d [tan x ] = sec2 x dx
d [tan (f (x ))] = sec2 (f (x ))f 0 (x ) dx
5. Secant
d [sec x ] = sec x tan x dx
d [sec (f (x ))] = sec (f (x )) tan (f (x ))f 0 (x ) dx
d [cot (f (x ))] = − csc2 (f (x ))f 0 (x ) dx d h (f (x )) i e = e (f (x )) f 0 (x ) dx d h (f (x )) i a = a(f (x )) ln af 0 (x ) dx
10. Natural Logarithm
d 1 [ln x ] = dx x
d 1 0 [ln f (x )] = f (x ) dx f (x )
11. Logarithm (base a)
d 1 [loga x ] = dx x ln a
d 1 [loga f (x )] = f 0 (x ) dx f (x ) ln a
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Summary of Derivative Rules
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Spring 2012
General Antiderivative Rules
Let F (x ) be any antiderivative of f (x ). That is, F 0 (x ) = f (x ). The most general antiderivative of f (x ) is then F (x ) + C . Original Function
General Antiderivative
1. Constant Rule
c (a constant)
cx + C
2. Constant Multiple Rule
cf (x )
cF (x ) + C
3. Sum Rule
f (x ) + g (x )
F (x ) + G (x ) + C
4. Difference Rule
f (x ) − g (x )
F (x ) − G (x ) + C
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Antiderivative Rules for Particular Functions Original Function
