Derivative Laws/Formulas

Power Rule

y^{}=x^{n}

y^{\prime}=nx^{^{n-1}}

(make the power a coefficient, and then subtract one from it, and put it back on top)

product rule

y=f\left(x\right)\cdot g\left(x\right)

y^{\prime}=f^{\prime}\left(x\right)\cdot g\left(x\right)+f\left(x\right)\cdot g^{\prime}\left(x\right)

identify two functions being multiplied, then derive both, multiply the derived version of one with the original of the other, do it vice versa, add the two

quotient rule

y=\frac{f\left(x\right)}{g\left(x\right)}

y^{\prime}=\frac{f^{\prime}\left(x\right)\cdot g\left(x\right)-f\left(x\right)\cdot g^{\prime}\left(x\right)}{\left(g\left(x\right)\right)^2}

product rule, but subtract the terms instead, and put over original second function, squared

chain rule

y=f\left(g\left(x\right)\right)

y^{\prime}=f^{\prime}\left(g\left(x\right)\right)\cdot g^{\prime}\left(x\right)

1. derive the outer, while maintain the inner (deadass just derive it and copy, don’t worry abt it yet)

2. then multiply by the derivative of the inner

Notes: never identify more than one layer at once. only look at the outermost and the immediate inner

also, EXPONENTS ARE OUTSIDE FUNCTIONS, NOT INNER

ln rule

y=\ln x

y^{\prime}=\frac{1}{x}

also note: \ln e=1

YOUVE ONLY FOUND THE DERIVATIVE OF THE OUTER HERE. CHAIN THE ARGUMENT IN

ln(g(x)) shortcut

y=\ln\left(g\left(x\right)\right)

y^{\prime}=\frac{g^{\prime}\left(x\right)}{g\left(x\right)}

because we always have to multiply the derivative of the inner function anyway, it always ends up in top of the original 1/g(x)

complex bases (anything with an x in its power)

y=a^{x}

y^{\prime}=a^{x}\cdot\ln a

so just the whole term, times ln(base)

note: if the power is itself a function, you’ve only found the derived outer (with inner intact, chain rule 1st step) so far. multiply by derived exponent

notes say know this:

you just gotta i guess

d/dx sinx =

cosx

d/dx cosx =

-sinx

d/dx tanx =

\sec^2x

d/dx secx =

sec\left(x\right)\cdot\tan\left(x\right)

log rule

y=\log_{a}x

y^{\prime}=\frac{1}{x\ln a}

once again, this is only the first step of chain rule. multiply by derivative of inner

to remember for physics

velocity func is derivative of displacement func

acceleration func is derivative of velocity func, or double derivative of displacement func

watch your signage on interval notations, the roots of the velocity graph are points where the object is at rest,

objects slow when the signs of their acceleration and velocity don’t match. they speed up when they do. make and compare two graphs to find this

logarithmic differentiation is just…

ln both sides, the y and the nasty func, and implicit diff

then use log laws to split it, typically quotient of log arguments into subtraction of two logs, exponent becoming coefficient rule, and multiplication of arguments becoming addition of logs

remember to plug back in

implicit diff shortcut

derive like the y is x, then multiply by dy/dx