# Differentiation

Here is a function.

If we draw tangent lines to the curve at some points,

we can see that where the tangent slopes up, the function increases,

and where it slopes down, the function decreases.

And where the tangent is a horizontal line, the function has a minimum,

but it also could have a maximum.

So, the tangent follows the movement of the function. If we compute the slope of the tangent line, that will tell us what happens

to the function itself.

Let's compute the slope of this tangent line, for instance.

The slope means how many units we step upwards while we step forward one unit.

To compute the slope we need to use another point.

First, we will compute the slope of the line

that goes through these two points.

Let's see what the slope of this line is!

moves up this much

moves forward this much

This slope is called the difference quotient, or Newton's quotient.

The slope of the secant is

the difference quotient:

This is truly splendid, but we originally wanted to compute the slope of the tangent.

Well, we will get to the tangent line by moving towards , such that the secants are gradually approaching the tangent.

The slope of the tangent is the limit of the slope of the secants.

This is called the differential quotient, or derivative by first principle.

The slope of the tangent is

the differential quotient:

at point it is the derivative

So, the derivative of a function tells us how steep of a tangent can be drawn to the curve of the function.

The derivative of function is denoted by .

Let's see the derivatives of some functions!

The derivative of a constant function is zero.

For example is a constant function, and

The derivative of power functions is

For example, the derivative of is

If we have to differentiate some radical thingy, that is done the same way:

and the derivative:

Function is an anchor point in our lives, as its derivative is itself:

The derivative of is a bit uglier:

For example, here is this one:

well, its derivative is not , because x is in the exponent.

This is an actual number, namely the natural logarithm of 5. There is no need to panic; we can compute it using a calculator:

This is splendid, but let's stick to .

And then here is the much talked-of and its derivative:

The derivative of other logarithms:

For example is base 10 logarithm, thus a=10, and the derivative:

And then we have the trigonometric functions.

The derivative of sine is cosine; the derivative of cosine is negative sine.

The derivative of tangent

well, that is a lot more unfriendly, let alone the others.

Now let's see the differentiation rules!

And here is the funniest: the differentiation rule for composite functions.

Here is a function, but this is not yet composite.

It will become a composite function if instead of x, we have something like

Now, this is a composite function, and the differentiation rule says that first we have to differentiate the outside function, which is

and then multiply it by the derivative of the inside function.

Here is another one.

This is not a composite function, only a harmless little sum.

But if this whole thing is raised to the fourth power,

then, well, it becomes a composite function.

The outside function is

and its derivative, as usual

and then it also has to be multiplied by the derivative of the inside function.

And here is this one, for instance.

The outside function

's derivative

Now it's time to try our luck

with some differentiation exercises.

This is not a composite function, only a harmless little sum.

But if this whole thing is raised to the fourth power,

then, well, it becomes a composite function.

Here is a function.

If we draw tangent lines to the curve at some points,

we can see that where the tangent slopes up, the function increases,

and where it slopes down, the function decreases.

And where the tangent is a horizontal line, the function has a minimum,

but it also could have a maximum.

So, the tangent follows the movement of the function. If we compute the slope of the tangent line, that will tell us what happens

to the function itself.

Let's compute the slope of this tangent line, for instance.

The slope means how many units we step upwards while we step forward one unit.

To compute the slope we need to use another point.

First, we will compute the slope of the line

that goes through these two points.

Let's see what the slope of this line is!

moves up this much

moves forward this much

This slope is called the difference quotient, or Newton's quotient.

The slope of the secant is

the difference quotient:

This is truly splendid, but we originally wanted to compute the slope of the tangent.

Well, we will get to the tangent line by moving towards , such that the secants are gradually approaching the tangent.

The slope of the tangent is the limit of the slope of the secants.

This is called the differential quotient, or derivative by first principle.

The slope of the tangent is

the differential quotient:

at point it is the derivative

So, the derivative of a function tells us how steep of a tangent can be drawn to the curve of the function.

The derivative of function is denoted by .

Let's see the derivatives of some functions!

The derivative of a constant function is zero.

For example is a constant function, and

The derivative of power functions is

For example, the derivative of is

If we have to differentiate some radical thingy, that is done the same way:

and the derivative:

Function is an anchor point in our lives, as its derivative is itself:

The derivative of is a bit uglier:

For example, here is this one:

well, its derivative is not , because x is in the exponent.

This is an actual number, namely the natural logarithm of 5. There is no need to panic; we can compute it using a calculator:

This is splendid, but let's stick to .

And then here is the much talked-of and its derivative:

The derivative of other logarithms:

For example is base 10 logarithm, thus a=10, and the derivative:

And then we have the trigonometric functions.

The derivative of sine is cosine; the derivative of cosine is negative sine.

The derivative of tangent

well, that is a lot more unfriendly, let alone the others.

Now let's see the differentiation rules!

And here is the funniest: the differentiation rule for composite functions.

Here is a function, but this is not yet composite.

It will become a composite function if instead of x, we have something like

Now, this is a composite function, and the differentiation rule says that first we have to differentiate the outside function, which is

and then multiply it by the derivative of the inside function.

Here is another one.

This is not a composite function, only a harmless little sum.

But if this whole thing is raised to the fourth power,

then, well, it becomes a composite function.

The outside function is

and its derivative, as usual

and then it also has to be multiplied by the derivative of the inside function.

And here is this one, for instance.

The outside function

's derivative

Now it's time to try our luck

with some differentiation exercises.

This is not a composite function, only a harmless little sum.

But if this whole thing is raised to the fourth power,

then, well, it becomes a composite function.