
a CONSTANT is zero.  
the PRODUCT OF A CONSTANT AND A FUNCTION is the product of the constant and the derivative of the function.  
the PRODUCT OF TWO FUNCTIONS, a first and a second function,
is the derivative of the first times the second plus the derivative of the second times the first.  
the QUOTIENT OF TWO FUNCTIONS, a first divided by a second function,
is the derivative of the first times the second minus the derivative of the second times the first, ALL divided by the square of the second.  
a FUNCTION OF A FUNCTION, a major function and its argument function, an outer
function and the inner function,
is the product of the derivative of the outer evaluated at the inner and the derivative of the inner. 
Arithmetic Tricks, Rules & Shortcuts In Words & Symbols 
The sine and cosine are closely related.
Take A Derivative or Antiderivative or Sine and Cosine 
Take First, Second, Third Derivatives of Sine & Cosine Functions GRAPHICALLY 
Use the graphs to takes the derivatives.
Use the stated derivatives (slopes) to describe the curve which is the derivative functions.
For example, TAKE THE DERIVATIVE OF THE SINE.
What's the function?
The cosine.
The derivative of the sine is the cosine.
Second Derivative
Two methods for taking the second derivative, the slope of the derivative, are suggested. EITHER repeat the above method using the cosine as the original function, OR, use the movement of the pencil point and the sine function to compute the second derivative of the sine, d^{2}[sin(x)]/dx^{2}.
To integrate a function times the derivative of another function, use:
