For derivatives, there are usually two ways of expressing a derivative:  dy/dx, or ƒ’.  I will be using ƒ’ for the sake of convenience (except for the first rule, so you can see how either way is usually expressed).

Note: For most rules I will not go into the derivation of each rule – the process would take entirely too long, and it is not necessary to know how each rule was made (even for AP AB calculus).

Letter representations:

c =  any constant

u and v =  any function

^n = any power of a variable.  So x² is the same thing as x^2.  The program I’m writing with won’t let me superscript anything higher than 3, so x^n is the best I can do.

‘ = first derivative (so u’ would mean the first derivative of function u)

Constant Rule:

dy/dx (c) = 0

or

ƒ’ (c) = 0

Example:  ƒ’ (4) = 0

Explanation: The letter c in calculus represents a constant (1, -5, 35, 134.67, etc).  So the derivative of any constant equal zero.  Why?  If you graphed y= -3, you would get a horizontal line.  Since the derivative tells us the rate of change of a function, a.k.a the slope of the function, and the slope of the line y=-3 (and all other constant functions), it stands to reason that the derivative (slope/rate of change) of those functions would always equal zero.  So the ƒ’(4) = 0.

The Constant Multiple Rule:

ƒ’(cu) = c×u’

Example:  ƒ’(4x) = 4 ׃’(x)

Explanation:  If you have the function 4x, and take the derivative of it, the derivative would be the derivative of x multiplied by the initial constant of 4.  So ƒ’(2x) = 2 × ƒ’(x).  But how do you take the derivative of a variable?  That’s where the Power Rule comes in.

The Power Rule:

ƒ’ (x^n) = n×x^(n-1)

Example:  ƒ’ (x³) = 3x²

Explanation:  To take the derivative of any variable, take the power of the derivative, n, and multiply it by the variable which is reduced by a power of one.  So ƒ’ (x^35) = 35x^34.  So ƒ’ (x) = 1×x^0, which equals 1.  But what if you have a quadratic function?  That’s where the Sum and Difference Rules come in.

Sum and Difference Rules

ƒ’(u ± v) = ƒ’(u) ± ƒ’(v)

Example:  ƒ’(x³ ± x²) = ƒ’(x³) ± ƒ’(x²)

Explanation:  The derivative of a sum is the sum of the derivatives.  The derivative of a difference is the difference of the derivatives.  So ƒ’(x³ + x²) = 3x² + 2x.

Now that we have our basic rules, let’s combine them for practice:

ƒ’(5x^6) = 30x^5  This combines the constant multiple rule and power rule.

ƒ’(3x^4 + 2x^8) = 12x^3 + 16x^7  This combines the constant multiple rule, the power rule, and the sum rule.

ƒ’(4x^2 + 5) = 8x  This rule combines all four rules detailed in this article.

But what if you had something like (5x^3 +5)(4x^13 + x^3 + 7) ?  Would want to foil that out first?  Or what if you had this?  For these you would need the product rule and quotient rule, which I will explain in my next article.

5x^3 + 2x + 3

    x^7 + 4

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