Derivatives - More Rules, Definitions, and Fundamentals

In this section we will largely be dealing with trigonometric and exponential functions.

D(f(x), f’(x) and d over dx derivative operator (f(x)) - The same thing!

A quick reminder, as we continue with derivatives is that derivatives of a function represented as D(f(x) or d over dx derivative operator (f(x)) or f’(x) mean the exact same thing.

In Math they are equally used throughout work and definitions, so do not concern yourself if you see them represented in different ways.

The Power Rule - for Functions

The Power Rule as we previously know it, just involving something to the power.

D(xn) = (n)(x(n-1))

Can also be applied in a similar manner when we have the entire function to a power,

Definition - Power Rule with Functions to a Power / The Power Rule for Functions

For a function, of say x, that is f(x), that is to the power of n, its derivative can be written as:

D(f(x)n) = (n)(f(n-1))D(f)

The Derivative of a function that is to the power n (a function that is being multiplied by itself n times) is equal ton times the function to the power of n - 1 times the derivative of the function.

This is different from the original Power Rule which deals with the derivative of some part of a function to a power, the Power Rule for Functions deals with the whole function to a power.

A Function to the power of something, can look like:

( f(x) )3, where f(x) is the function and n = 3

It can also be seen as examples like:

x5, where x is the function and n = 5

(2x3 + 3x+ 5)7, where 2x3 + 3x+ 5 is the function and n = 7

An example that would not be true is

(x2 + 2) thinking that (x + 2) is the function and n = 2, this would be incorrect, the whole function must be to the power of the same thing.

We will learn to identify derivatives that have the whole function to the power of something, then we can use the power rule on it.

Beginner Power Rule for Functions

Use the Power Rule to calculate the derivatives of the following functions:

a) d over dx derivative operator(f(x)2)

First, recall the Power Rule:

d over dx derivative operator(f(x)n) = n.f(x)n-1.d over dx derivative operator(f(x))

For d over dx derivative operator(f(x)2) we have f(x) = f(x) and n = 2

Note: When we have questions with general non-specific terms like f(x), it is expected that they are left in that form. You don't have to worry about what f(x) equals exactly.

d over dx derivative operator(f(x)2) = 2.f(x)2-1.d over dx derivative operator(f(x))

= 2.f(x).d over dx derivative operator(f(x))


b) d over dx derivative operator(x3)

Here we may see x3 as a term itself, but with the Product Rule for Functions, we can start to identify functions within functions. x3 is a function but x itself is also a function. So we can see x to the power of 3 as a function to the power of 3.

So recall the Power Rule:

d over dx derivative operator(f(x)n) = n.f(x)n-1.d over dx derivative operator(f(x))

For d over dx derivative operator(x3) we have f(x) = x and n = 3

As we can see, x itself can be a function.

So, d over dx derivative operator(x3) = 3.x3-1.d over dx derivative operator(x)

= 3x2.(1)

= 3x2

Note how we can also get this answer using the standard Power Rule method.

More Power Rule for Functions Examples

Trigonometric Functions - Derivatives

To remind ourselves, we know that:

D(sin(x)) = cos(x)

D(cos(x)) = -sin(x)

Below are some more derivatives of trigonometric functions that will be useful in the future.

Definitions - Trigonometric Derivatives - tan(x), sec(x), csc(x) and cot(x)

Here we will introduce more derivatives of trigonometric functions.

D(tan(x)) = sec2(x)

D(sec(x)) = sec(x)tan(x)

D(cosec(x)) or D(csc(x)) = -csc(x)cot(x)

D(cot(x)) = -csc2(x)

These derivatives can be proved by first writing each of them in their alternative formats, then using the quotient rule to differentiate them.

Beginner Trigonometric Derivatives

Calculate the derivatives of the Trigonometric functions below.

a) d over dx derivative operator(2tan(x))

= 2d over dx derivative operator(tan(x)) , (Constant Multiple Rule)

= 2(sec2(x)) , (From Trigonometric Derivative Rules)


a) D(15sec(x))

= 15(D(sec(x)))

= 15(sec(x)tan(x))


c) d over da derivative operator(3cosec(a))

= 3d over da derivative operator(cosec(a))

= 3(-cosec(a)tan(a))

= -3cosec(a)tan(a)


d) d over dt derivative operator(3cot(t))

= 3d over dt derivative operator(cot(t))

= 3(-cosec2(t))

= -3cosec2(t)

More Derivatives of Trigonometric Functions Examples

Derivatives of ex

Definition - Derivative of Exponentials - ex

D(ex) = ex

Derivatives with ex - Power Rule for Functions Product Rule and Quotient Rule

Calculate the following derivatives below that involve ex

The solution to finding the derivative of (x times e^x)
The solution to finding the derivative of e^x divided by cos(x)

The solution to finding the derivative of e^(3x)

Differentiating a Derivative - Higher Derivatives

We know that a derivative of a function represents the equation for the slope of that function at a given point, it can also represent things like rates of change or acceleration. Now we will start to look at derivatives of derivatives. This will mean looking at what the slope of a derivative is or things like how fast is the rate of change changing.

For example, if we have an equation that represents the speed of something, for t as time,

speed = 3t3

Then the acceleration would be the derivative of that,

Acceleration = 9t2

Definition - First, Second and Third Derivatives

For a function with y = f(x)

The first derivative is,

f’(x)

Or

d over dx derivative operator


The second derivative is

D(f’(x)) = f’’(x)

Or

d over dx derivative operator opening bracket dy over dx derivative operator closing bracket  =  second derivative operator


The third derivative is

D(f’’(x)) = f’’’(x)

Or

d over dx derivative operator opening bracket second derivative operator closing bracket  =  third derivative operator

More Higher Derivatives Examples

Lesson Formulas

Power Rule for Functions

D( f(x)n ) = n (f(x)(n-1)) D(f(x))

Exponential / e Derivative

D(ex) = ex

Trigonometric Function Derivatives

D(tan(x)) = sec2(x)

D(sec(x)) = sec(x)tan(x)

D(cosec(x)) or D(csc(x)) = -csc(x)cot(x)

D(cot(x)) = -csc2(x)