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 x^{n} is n*x^{n-1}.

Ex [2] ^{dy}/_{dx}
3x^{2} = 6x.

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

Ex [3] ^{dy}/_{dx
}6x^{3} + 4x^{2} + 8x - 4 = _________

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

b. So this
becomes ^{dy}/_{dx }6x^{3} + ^{dy}/_{dx}
4x^{2} + ^{dy}/_{dx }8x - ^{dy}/_{dx} 4
= 18x^{2} + 8x + 8 - 0.

c. The
answer is 18x^{2} + 8x + 8.

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

Ex [4] ^{dy}/_{dx}
(3x^{2}+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}
3x^{2} + 5x + 2 = 6x + 5. This represents f '(x).

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

d. Now, multiplying these two values together gives:

(6x+5)*3*(3x^{2}+5x+2)^{2}
or (18x+15)(3x^{2}+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)(x^{2}-3) = _________

a. First,
multiply the derivative of the first times the second. So we get: ^{dy}/_{dx}
3x - 4 = 3. So 3(x^{2}-3) is the first term.

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

c. The
answer is 3(x^{2}-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)*sec^{2}(f(x)).

4. The
derivative of e^{f(x)} = f '(x)*e^{f(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(2^{2}) = 12.

c. The answer is 12.

Ex [3] If
f(x) = 4x^{3} - 12x^{2} + 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) =
12x^{2} - 24x + 4

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

d. The answer is 24.