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Derivatives

PostPosted: August 24th, 2011, 1:04 pm
by Jessica
The derivative of Image with respect to x is the function Image and is defined as,
Image



Example
Image


First plug the function into the definition of the derivative.
Image

Multiply everything out
Image

Image

So the derivative is
Image

Re: Derivatives

PostPosted: October 18th, 2011, 10:08 pm
by Jessica
this is the alternate definition of the derivative.


Re: Derivatives

PostPosted: October 23rd, 2011, 9:01 pm
by Jessica

Re: Derivatives

PostPosted: October 23rd, 2011, 9:02 pm
by Jessica
Rule 1: Derivative of a Constant

If c is a constant, then




Rule 2: Power Rule for a Positive Integer Powers of x

If n is a positive integer, then nxn−1




Rule 3: The Constant Multiple Rule

If u is a differentiable function of x and c is a constant, then




Rule 4: The Sum and Difference Rule

If u and v are differentiable functions of x, then their sum and differences are differentiable at every point where u and v are both differentiable. At such points,

1.


2.




Rule 5: The Product Rule

If the products of two differentiable functions u and v is differentiable and...

    u = first factorv = second factor




(first) · (second)' + (second) · (first)'



Finding Horizontal Tangents:

Horizontal tangents occur when the slope of the derivative equals zero.





Rule 6: The Quotient Rule

At a point where v ≠ 0, the quotient of two differentiable functions is differentiable and...

u = numerator part of quotient
v = denominator part of quotient



Re: Derivatives

PostPosted: November 12th, 2011, 12:39 pm
by Jessica
Derivatives of the Six Trig Functions












Re: Derivatives

PostPosted: November 16th, 2011, 12:26 am
by Jessica
Chain Rule

useful when asked to find the derivative of compositions such as


Method 1: "outside-inside" rule

   •    ( 2 )
deriv of outside      deriv of inside
leave inside alone





Method 2:

Examples coming soon

Re: Derivatives

PostPosted: November 19th, 2011, 10:09 pm
by Jessica
Finding parametrically:


Re: Derivatives

PostPosted: December 5th, 2011, 5:05 pm
by Jessica
Derivatives of Inverse Trig Functions



is exactly the same except it's negative.




is exactly the same except it's negative.




is exactly the same except it's negative.



Image

Re: Derivatives

PostPosted: December 8th, 2011, 8:42 am
by Jessica
Image

Image

note: each will be multiplied by