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A.  A derivative is calculated the exact opposite to that of an integral

B.  A function's derivative is basically the equation for the slope of the original function.  Derivatives are usually expressed by: f '(x) or y' or dy/dx.  f '(x) or y' is the first derivative.  f ''(x) or y'' is the second derivative and so on.

C.  Below are the basic rules for computing derivatives.

1.  The derivative of a constant is 0.

Ex [1]  dy/dx 5 = 0.

2.  The derivative of xn is n*xn-1.

Ex [2]  dy/dx 3x2 = 6x.

3.  The derivative of f(x) g(x) = f '(x) g'(x)

Ex [3]  dy/dx 6x3 + 4x2 + 8x - 4 = _________

a.  This rule means you can take each term separately.

b.  So this becomes dy/dx 6x3 + dy/dx 4x2 + dy/dx 8x - dy/dx 4 = 18x2 + 8x + 8 - 0.

c.  The answer is 18x2 + 8x + 8.

4.  The derivative of [f(x)]n is f '(x)*[f(x)]n-1.

Ex [4]  dy/dx (3x2+5x+2)3 = ________

a.  You always want to work from the inside out.

b.  The first step is to take the derivative of the inside first.  So dy/dx 3x2 + 5x + 2 = 6x + 5.  This represents f '(x).

c.  Now, we need the derivative of the outside which is 3(3x2+5x+2)2.

d.  Now, multiplying these two values together gives: 

(6x+5)*3*(3x2+5x+2)2 or (18x+15)(3x2+5x+2)2.

5.  The derivative of f(x)*g(x) = f '(x)g(x)+f(x)g'(x).

a.  This type of problem will probably not be found on a number sense test.

Ex [5]  dy/dx (3x-4)(x2-3) = _________

a.  First, multiply the derivative of the first times the second.  So we get: dy/dx 3x - 4 = 3.  So 3(x2-3) is the first term.

b.  Next, multiply the first term times the derivative of the second.  So we get: dy/dx x2 - 3 = 2x.  So we get 2x(3x-4) for the second term.

c.  The answer is 3(x2-3)+2x(3x-4).

D.  Common Derivatives:

1.  The derivative of sin(f(x)) = f '(x)*cos(f(x)).

2.  The derivative of cos(f(x)) = -f '(x)*sin(f(x)).

3.  The derivative of tan(f(x)) = f '(x)*sec2(f(x)).

4.  The derivative of ef(x) = f '(x)*ef(x).

5.  The derivative of ln (f(x)) = f '(x)*1/f(x).

E.  Examples

Ex [1]  If f(x) = sin 4x, then f '() = _______

a.  First, take the derivative of 4x which is 4.

b.  The derivative of sin x is cos x, so the derivative of sin 4x = 4 cos 4x. 

c.  Plugging in x = , we get 4 cos = 4(-1) = -4.

d.  The answer is -4.

Ex [2]  Find the slope of the tangent line of y = (x-2)3 at the point x = 4.

a.  Since a derivative is the slope of the tangent line, we need to find y' and use the value x = 4 to find the slope.

b.  y' = 3(x-2)2, so plugging in x=4 we get 3(22) = 12.

c.  The answer is 12.

Ex [3]  If f(x) = 4x3 - 12x2 + 4x - 3, then f ''(2) = ______

a.  For this problem we are looking for the second derivative.  So we need to find the derivative of the derivative, or just take the derivative twice.

b.  f '(x) = 12x2 - 24x + 4

c.  So f ''(x) = 24x - 24.  f ''(2) = 24(2) - 24 = 24.

d.  The answer is 24.


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