Derivatives Cheat Sheet

Derivatives Cheat Sheet Derivative Rules 1. Constant Rule: 2. Power Rule: d (c)=0, where c is a constant dx d n (x )=nxn−1 dx 3. Product Rule: (f g)′ = f ′ g + f g′ ′ f ′ g −gf2 g′ f = 4. Quotient Rule: g 5. Chain Rule: (f (g (x))′ = f ′ (g (x))g ′ (x) Common Derivatives Trigonometric Functions d (cosx)=− sinx dx d (sinx) = cosx dx d (secx) = secx tanx dx d dx d dx (cscx)=− cscx cotx d dx (tanx)=sec2 x (cotx)=− csc2 x Inverse Trigonometric Functions d (sin−1 x)= dx 1 √ 1− x2 1 √ 1 − x2 d (cos−1 x)=− dx d (tan−1 x)= dx Exponential & Logarithmic Functions d d x x (a )=a ln(a) dx d (loga (x))= dx dx 1 x ln(a) 1 (ex )=ex 1 d (ln(x))= x dx 1 1+x2 Chain Rule In the below, u = f (x) is a function of x. These rules are all generalizations of the above rules using the chain rule. 1. du d n (u )=nun−1 dx dx 2. d u u du (a )=a ln(a) dx dx 3. d u u du (e )=e dx dx 4. d (loga (u))= dx 5. 1 du d (ln(u))= dx u dx 6. du d (sin(u))=cos(u) dx dx 7. d du (cos(u))=− sin(u) dx dx 8. du d (tan(u))=sec2 (u) dx dx 1 du x ln(u) dx 9. Same idea for all other trig functions 10. d (tan−1 (u))= dx 1 1+u2 du dx 11. Same idea for all other inverse trig functions Implicit Differentiation Use whenever you need to take the derivative of a function that is implicitly defined (not solved for y ). Examples of implicit functions: ln(y)=x2 ,x3 + y 2 =5, 6xy=6x +2y 2 , etc. Implicit Differentiation Steps: 1. Differentiate both sides of the equation with respect to “ x” 2. When taking the derivative of any term that has a “y” in it multiply the term by y ′ (or dy/dx) 3. Solve for y ′ When finding the second derivative y ′′ , remember to replace any y ′ terms in your final answer with the equation for y ′ you already found. In other words, your final answer should not have any y ′ terms in it. 2 Log Differentiation Two cases when this method is used: • Use whenever you can take advantage of log laws to make a hard problem easier – Examples: (x3 + x)cosx , ln(x2 + 1)cos(x)tan−1 (x), etc. x2 +1 – Note that in the above examples, log differentiation is not required but makes taking the derivative easier (allows you to avoid using multiple product and quotient rules) • Use whenever you are trying to differentiate d  dx f (x)g(x)  √ – Examples: xx ,x x , (x2 +1)x , etc. – Note that in the above examples, log differentiation is required. There is no other way to take these derivatives. Log Differentiation Steps: 1. Take the lnof both sides 2. Simplify the problem using log laws 3. Take the derivative of both sides (implicit differentiation) 4. Solve for y ′ 3