Who can give me a crash course into derivatives? I already know Limits and Conti

Question

Who can give me a crash course into derivatives? I already know Limits and Continuity, just would like to know the basic rules of derivations and an easily understandable example of each one. Who can give me a crash course into derivatives? I already know Limits and Continuity, just would like to know the basic rules of derivations and an easily understandable example of each one. Who can give me a crash course into derivatives? I already know Limits and Continuity, just would like to know the basic rules of derivations and an easily understandable example of each one.

Explanation / Answer

I can try : ) i'm new at answering question so hopefully this helps

derivatives are basically rates of change. the derivative of accleration, for example, is velocity. deriving velocity will give you its position. this is the most common way of seeing what derivatives can give you.

but as for just solving standard derivatives here are a couple of the rules. i do not know how far your class is or what techs you know.

power rule: your absolute best friend, when you can use power rule life is easy: d/dx xn = nxn-1 you just bring the exponent down front and multiply by it and reduce the exponent by 1.

EX: y = x2 + x5      sol: y' = 2x + 5x4         EX 2: y = 1/x3 which can be rewritten as y= x-3 sol: y' = -3x-4 or -3/x4

but what if you have just y=x? if x is to the power of one, its derivative is just the constant. if y = 3x, y'=3

if you derive a constant, it goes to 0. d/dx (5) = 0

now that the basics are out of the way, what about polynomials?

you have two rules you must know. the product rule and the quoitent rule.

the product rule: best explained by example: y= (2x3+x+5)*(x2 +5x+2)    so, the product rule states the derivative is equal to the derivative of the 1st term, times the underived second term plus the derivative of the second term times the underived first term. that is [d/dx(first)*(second)] + d/dx(second)*(first)].      in this example    y' = (6x2+1)*(x2 +5x+2) + (2x+5)*(2x3 +x+5)   this is the derivative and then you can simplify through algebra if needed or leave it as is if your teacher is not picky.

quoitent rule is similar but a bit "harder". say we have (x2 +5) / (x3 + 2x).    quoitent rule states the derivative of a polynomial fraction is the derivative of the top term times the bottom term minus the derivative of the bottom term times the top term, ALL over the bottom term squared. that is [d/dx(top)*(bottom) - d/dx(bottom)*(top)] / (bottom)2

solution: [(2x)*(x3 +2x) - (3x+2)*(x2 +5)] / (x3 + 2x)2 again simplify as needed using algebra

there is also chain rule which deals with functions inside of functions. that is such as y = sin(2x5 +3)

chain rule states you derive the outside function first then multiply by the derivative of the inside.

solution: cos(2x5+3) * 10x4

that covers a lot, does that make sense? feel free to ask questions.

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