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

**D**(f(x), f’(x) and (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 (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 as we previously know it, just involving something to the power.

**D**(x^{n}) = (n)(x^{(n-1)})

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

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 to** →*n* **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:

x^{5}, where x is the function and n = 5

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

An example that would **not** be true is

(x^{2} + 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.

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

a) (f(x)^{2})

First, recall the Power Rule:

(f(x)^{n}) = n.f(x)^{n-1}.(f(x))

For (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.

(f(x)^{2}) = 2.f(x)^{2-1}.(f(x))

= 2.f(x).(f(x))

b) (x^{3})

Here we may see *x*^{3} as a term itself, but with the Product Rule for Functions, we can start to identify functions within functions. *x*^{3} 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:

(f(x)^{n}) = n.f(x)^{n-1}.(f(x))

For (x^{3}) we have f(x) = x and n = 3

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

So, (x^{3}) = 3.x^{3-1}.(x)

= 3x^{2}.(1)

= 3x^{2}

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

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.

Here we will introduce more derivatives of trigonometric functions.

**D**(tan(x)) = sec^{2}(x)

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

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

**D**(cot(x)) = -csc^{2}(x)

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

Calculate the derivatives of the Trigonometric functions below.

a) (2tan(x))

= 2(tan(x)) , (Constant Multiple Rule)

= 2(sec^{2}(x)) , (From Trigonometric Derivative Rules)

a) D(15sec(x))

= 15(D(sec(x)))

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

c) (3cosec(a))

= 3(cosec(a))

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

= -3cosec(a)tan(a)

d) (3cot(t))

= 3(cot(t))

= 3(-cosec^{2}(t))

= -3cosec^{2}(t)

Definition - Derivative of Exponentials - e^{x}

**D**(e^{x}) = e^{x}

Calculate the following derivatives below that involve e^{x}

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 = 3t^{3}

Then the acceleration would be the derivative of that,

Acceleration = 9t^{2}

For a function with y = f(x)

The first derivative is,

f’(x)

Or

The second derivative is

**D**(f’(x)) = f’’(x)

Or

=

The third derivative is

**D**(f’’(x)) = f’’’(x)

Or

=

D( f(x)^{n} ) = n (f(x)^{(n-1)}) D(f(x))

D(e^{x}) = e^{x}

**D**(tan(x)) = sec^{2}(x)

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

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

**D**(cot(x)) = -csc^{2}(x)