in this video we're going to go over some differentiation formulas particularly if you're studying derivatives in calculus so if you have a sheet of paper with you feel free to get ready to take down some notes so the first thing you want to be familiar with is the derivative of a constant the derivative of a constant is always a zero the next Formula you need to know is the power rule or the derivative of a power function here we have a variable raised to a constant it's going to be that constant times the variable raised to the N minus 1. so for instance the derivative of x cubed is 3x squared the derivative of x to the fourth is 4X cubed the derivative of x to the fifth is five x to the fourth so that's how you can employ the power rule to find the derivatives of functions like this now instead of having a variable raised to a constant what if we have a constant raised to a variable the derivative of a to the x is going to be a to the x times Ln a now the reason why we get that is because the derivative of x is one but let's say we have the derivative of a to the U where a is a constant but U is a function of x this is going to be a to the U times the derivative of U times Ln a here we would have the derivative of x but that's just going to be 1. now if you ever get this the derivative of a variable raised to a variable this rather than having a formula you need to employ a process called logarithmic differentiation and I have a video on YouTube that covers that so if you go to the YouTube search bar and type in logarithmic differentiation organic chemistry tutor you should see a video that will come up and explain how to do problems like this next up we have something called the constant multiple rule so if you have if you're trying to find the derivative of a function multiplied by some constant C it's simply going to be that constant times the derivative of that function so for instance let's say if we want to find the derivative of 5x to the 4. we know the derivative of x to the 4 but we can rewrite this as 5. times the derivative of x to the fourth and using the power rule we know this is going to be 5 times 4X cubed which becomes 20x cubed the next Formula you need to be familiar with is the power rule so if we have two functions u and v and they're multiplied to each other the derivative of the product of these two functions is going to be U Prime V plus UV Prime so it's the derivative of U times V plus u times the derivative of V next up we have the quotient rule so here we have a fraction of two functions or a division of two functions and it's going to be v u Prime minus u v Prime over V squared so that's the formula associated with the quotient rule now sometimes you may need to find the derivative of a composite function in this case you need to use a chain rule so let's say we have f of G of U we want to find the derivative of that so first we're going to find the derivative of the outer function f we're going to keep the inside part the same and then we're going to multiply by the derivative of the inside part that is the derivative of G and then we'll multiply by the derivative of the inside of G which is U so that will be times U Prime now you might see the chain rule represented as a function of X instead of U so let's say if you have this F of G of x it's going to be the derivative of the outside function we'll keep the inside part the same and then times the derivative of the inside function now the derivative of x is one so there's no point writing that it would just be times one so if you have X this is all you need but if you have a function U or a function like where use a function of X you're also going to have U Prime at the end so keep that in mind so you got to differentiate the outer function f and then work your way towards the middle then G and then U now for those of you who want additional example problems on this check out the links in the description section below now let's continue let's talk about another form of the chain rule when it's combined with the power rule so let's say we want to find the derivative of the function f of x but it's raised to the n so first we're going to focus on the outside part we're going to keep the inside part the same so it's going to be n times f of x to the N raised to the N minus 1. much like the power rule where was n X raised to the N minus 1. but we do have a function on the inside it's not just X it's a function of X so now we've got to find the derivative of the inside so we're going to multiply it by the derivative of the inside so this combines the power rule with a chain rule another formula that's associated with the chain rule is this one d y d x is equal to d y d u times d u over DX now let's talk about the derivative of logarithmic functions so let's say we want to find the derivative of log base a of U where U is a function of x it's going to be U Prime over U LNA if we want to find the derivative of the natural log of U keep in mind the base of a natural log of E it's going to be U Prime over U it's the same as this one the only issue is Ln e is equal to one so you could write it as just U on the bottom so those are the two formulas you need to be familiar with when finding the derivatives of logarithmic functions now let's focus on trig functions the derivative of sine of U is going to be cosine U times U Prime now if you just have sine X the derivative of sine X is simply cosine X you could think of it as cosine X and the derivative of x is one so it's just cosine X but let's say if you was x squared if you want to find the derivative of sine X Squared it's going to be cosine x squared and then times the derivative of x squared which would be 2X so this is the U part and this is the U Prime part that's why I like to write it in this format it reminds you that you'll need to employ the chain rule if you have something other than x as the angle now the derivative of cosine U this is going to be negative sine U times U Prime the derivative of tangent of U is secant squared times U Prime now for cotangent it's going to be negative cosecant squared and as always times U Prime now the next two that you need to know are the derivative of secant and the derivative of secant's cousin or cosecant and they're quite similar if there's a c in front typically it's going to have a negative sign the derivative of secant is you know what let's change this let's change it from X to a u the derivative of secant U is going to be secant U tangent U times U Prime the derivative of cosecant use negative cosecant U cotangent U times U Prime now the next set of formos need to be familiar with are the inverse trig formulas so let's start with the derivative of the inverse of sine of U so that's U Prime over the square root of 1 minus U squared now just for comparison purposes if you have the inverse sine of x it's going to be 1 over the square root of 1 minus x squared because the derivative of x is one you're going to have that there but if you have U you can have U Prime instead of one so make sure you're mindful of that difference now the derivative for the inverse cosine of U is going to be very similar to the derivative of sine the only difference is it's going to have a negative sign but everything else is going to be the same now let's move to the arc tangent function so the derivative of inverse tan of U it's going to be U Prime over 1 plus u squared now for Arc cotangent it's going to be negative U Prime over 1 plus u squared next up we have inverse secant and the formula for that is going to be U Prime over U square root U squared minus 1. and for inverse cosecant it's negative U Prime over U square root U squared minus 1. so those are the formulas for the derivatives of the inverse trig functions so that's it for this video so if you're studying for a derivatives test at least you know the most common formula is that you'll need and that you can be tested on so hopefully you wrote those down and thanks for watching