Derivative Help(Calculus)

The derivative by definition
lim?x?0fx+?x-fx?x

lawsuit:
fx=x2
fx+?x-fx?x
(x+?x)2-x2?x
x2+2x?x+?x2-x2?x The x2 cancel out because on is + and the other is -
2x?x+(?x)2?x
?x2x+?x?x set up the equation. The ?x will cancel out because you are dividing the homogeneous thing.
2x+?x
2x in the answer because ?x will go to 0 which leaves 2x.

first derivative of a product
y=gx?hx
Or
y=u?v
dydx=udvdx+vdudx

Example:
ddx4x+37x-1
dydx=udvdx+vdudx
4x+3dvdx7x-1+7x-1dudx4x+3
4x+37+7x-1(4) multiply the 7 into the 4x+3 and the 4 into the 7x-1.
28x+21+28x-4 match or set forth like terms.
56x+17 in the answer.

Derivative of a quotient
y=g(x)h(x)
Or
y=uv
ddxuv=vdudx-udvdxv2

Example:
ddx5x+34x2-7
ddxuv=vdudx-udvdxv2
4x2-7dudx5x+3-5x+3dvdx4x2-74x2-72
4x2-75-5x+38x4x2-72 multiply the 5 into the 4x2-7 and the 8x into the 5x+3.
20x2-35-40x2-24x4x2-72 add or subtract like terms.
-20x2-24x-354x2-72 is the answer.

The Derivative of a power
dydx=ddxun=nun-1dudx

Example:
y=3x+250
dydx=ddx3x+250
503x+250-1ddx(3x-2)
503x+249(3) multiply the 3 and 50
150(3x+2)49 is the answer

The chain rule
dydx=dydx?dudx

Example:
y=3x2+5
dydx=ddx3x2+5
ddx3x2+512
123x2+512-1ddx3(2)x2-1+5
123x2+5-126x multiply 6x into 12.




3x3x2+5 is the answer

inherent Differentiation

Example:
dydx if 2y3+3x2y-8x3+2x-y=dydx0
23y3-1+3x2y1-1+y32x2-1-83x3-1+21x1-1-y1-1dydx=0
6y2+3x2+6xy-24x2+2-1dydx=0
6y2+3x2-1dydx=-6xy+24x2-2
dydx=-6xy+24x2-26y2+3x2-1

Velocity: v=dsdt (s is the rate change and t is the time)
Acceleration: a=dvdt=d2sdt2

Derivative of transcendental functions
ddx(sinu)=cosududx
ddxcosu=-sinududx
ddxtanu=sec2ududx
ddxcotu=-csc2ududx
ddxsecu=secutanududx
ddxcscu=-cscucotududx

Inverse trigonometric functions
ddxacrsin u=11-u2dudx
ddxarccosu=-11-u2dudx
ddxarctanu=11+u2dudx
ddxarccotu=-11+u2dudx
ddxarcsecu=1u2u2-1dudx
ddxarccscu=-1u2u2-1dudx

Laws of exponents
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