nick zurawski jon stensberg some other kid. everyone hates formal definition of a derivative, so...
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Power Rule Everyone hates
formal definition of a derivative, so use this instead
d/dx (x^n) = nx^(n-1)
Bring it down, subtract one from the power
Y=X3+X Do each piece separately 3X2
1 Y’=3X2+1
A=πr2
A’=2πr Should look familiar… Circumference is just the power rule
derivative of Area
Y=4X8/3-6X7/2+2X1/8
Solve each part 32/3 * X5/3
21 * X5/2
¼ * X-7/8
Y’= 32/3*X5/3 - 21*X5/2 + 1/4*X-7/8
Product Rule• LITERALLY the most forgotten rule in
Calculus. If you forget about the product rule, Ms. Scarseth will incessantly mock you in front of your peers – learn it!
• RULE: First term multiplied by the derivative of the second term PLUS the second term multiplied by the derivative of the first term.
Producto Rulo Examplos Y=(x+1)(x-1) Y’=(1)(x-1) + (x+1)(1) Y’=x-1+x+1 Y’=2x Check! Y=x2-1 Power Rule Y’=2x Correctomundo!
Y=(x-1)(x3+2x-1) Y’=(1)(x3+2x-1) + (x-1)(3x2+2) Y’=x3+2x-1+3x3+2x-3x2-2 Y’=4x3-3x2+4x-3
Y=(x4+5x3)(5x2+2)(7x-1+4) Y’=a’bc+ab’c+abc’ Y’= (4x3+15x2)(5x2+2)(7x-1+4)
+(x4+5x3)(10x)(7x-1+4)+(x4+5x3)(5x2+2)(-7x-2)
Smart people algebra stuff happens Y’=120x5+675x4+732x3+162x2+140
x
Chain Rule Kinda like the power rule, but not
really so hold onto your horses
d/dx f(g(x)) = f'(g(x))g'(x)
Derive the outside thingy and keep the inside thingy the same and then multiply by the derivative of the inside thingy.
#YoloChain Y=4tan(x2) Y’=8x*sec2(x2)
Y=(cot(x)+sin(x))2
Y’=2(cot(x)+sin(x)) *(-csc2(x)+cos(x))
Y=1/6*tan(esec(x)) Move through each piece outside-in 1/6 sec2(esec(x)) esec(x)
sec(x)tan(x) Y’=1/6 *sec2(esec(x)) *esec(x)
*sec(x)tan(x)
Quotient Rule You’re going to hate this, sorry
Low D High minus High D Low over Bottom^2… but if that’s not good enough for you, mister fancy math pants, then here
f(x) = g(x)/h(x)
f’(x) = [g’(x)h(x)-g(x)h’(x)]/[h(x)]^2
Quotient Rule For Life Y=(2x+5)/(3x-2)
Y’=[(2)(3x-2)-(2x+5)(3)]/(3x-2)2
Y’=-19/(3x-2)2
Y=(x2+x)/(x2-x) Y’=[(2x+1)(x2-x)-(x2+x)(2x-1)]/(x2-x)2
Y’=[2x3-x2-x-2x2-x2+x]/[x4-2x3+x2] Y’=[-2x2]/[x4-2x3+x2] Y’=-2/[x2-2x+1]
Resources Calculus: Graphical, Numerical, Algebraic by
Finney, Demana, Waits, and Kennedy
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