Essential Knowledge 2.1C2

Essential Knowledge 2.1C2 Students will know that specific rules can be used to calculate derivatives for classes of functions, including polynomial, rational, power, exponential, logarithmic, trigonometric, and inverse trigonometric.

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When it comes to improving your ability to perform calculus, this might be the most important topic. These are the shortcuts for doing derivatives, and you will always want to use these. Not only do these dramatically improve your ability to determine derivatives, they also improve your ability to perform integrals. The reason is, since integrals are the opposites of derivatives, the shortcuts for integrals are just the opposite of the shortcuts for derivatives.

You could probably Google these and find them in a thousand different websites, since they are so important. This website came up first when I Google'd "differentiation rules". I'll put them here anyways.

Coefficient Rule

This just means if you want to use any rule, you can just ignore any coefficients, use the rule, then multiply the coefficients back in after.

Power Functions

The rule you'll use the most. Whatever the exponent of the original function was, your derivative will have that as a coefficient. Whatever the exponent of the original function was, your derivative will have that exponent minus 1. Examples make this more clear.

used the coefficient rule, (3)(2)=6

used the coefficient rule, (-5)(1/2)=-5/2

Polynomial Functions

Just use the power rule on each term

Rational Functions

A rational function is a polynomial divided by a polynomial. Since you are dividing, these require a special division rule called the "quotient rule". You'll learn the quotient rule in the next Essential Knowledge. But other than that, you just use the power rule on each term, just like polynomials.

Exponential Functions

In other words to find the derivative, take the original function, and multiply it by the natural log of the base.

Notice something special would happen for bases of e. (And the base will almost always be e, by the way.)

Logarithmic Functions

In other words, change the base of the log becomes e, it goes in the denominator, and gets multiplied by x.

Notice something special would happen for bases of e. (And the base will almost always be e, by the way.) Remember, logs with base e are called "natural logs" and change to "ln". In other words, remember that ln(x) = log_e(x)

Trigonometric Functions

Sine, cosine, tangent, cosecant, secant, and cotangent all have derivatives. You should memorize them, especially sin(x) and cos(x)

Here they are

Inverse Trigonometric Functions

Inverse sine, inverse cosine, inverse tangent, inverse cosecant, inverse secant, and inverse cotangent all have derivatives. I've never memorized these. I don't think you should bother honestly

Here they are