5applications Of The Derivative Part 2ap Calculus

Posted : admin | On 30-03-2021

1 Lecture 2.4: Applications of the Derivative. To find the derivative of this function, we need to use the Fundemental Theorem of Calculus Part 1 (As opposed to the 2nd part, which is what's usually used to evaluate definite integrals). Take derivatives of both sides. 'Cancel' the integral and the derivative.

It is all about slope!

Slope = Change in YChange in X

We can find an average slope between two points.

But how do we find the slope at a point?

There is nothing to measure!

But with derivatives we use a small difference ...

... then have it shrink towards zero.

Let us Find a Derivative!

To find the derivative of a function y = f(x) we use the slope formula:

Slope = Change in YChange in X = ΔyΔx

And (from the diagram) we see that:

x changes fromxtox+Δx
y changes fromf(x)tof(x+Δx)

Now follow these steps:

  • Fill in this slope formula: ΔyΔx = f(x+Δx) − f(x)Δx
  • Simplify it as best we can
  • Then make Δx shrink towards zero.

Like this:

Example: the function f(x) = x2

We know f(x) = x2, and we can calculate f(x+Δx) :

Start with:f(x+Δx) = (x+Δx)2
Expand (x + Δx)2: f(x+Δx) = x2 + 2x Δx + (Δx)2
Put in f(x+Δx) and f(x):x2 + 2x Δx + (Δx)2 − x2Δx
Simplify more (divide through by Δx):= 2x + Δx

Result: the derivative of x2 is 2x

In other words, the slope at x is 2x

We write dx instead of 'Δx heads towards 0'.

And 'the derivative of' is commonly written :

x2 = 2x
'The derivative of x2 equals 2x'
or simply 'd dx of x2 equals 2x'

5 Applications Of The Derivative Part 2ap Calculus Answers

What does x2 = 2x mean?

It means that, for the function x2, the slope or 'rate of change' at any point is 2x.

So when x=2 the slope is 2x = 4, as shown here:

Or when x=5 the slope is 2x = 10, and so on.

Note: sometimes f’(x) is also used for 'the derivative of':

f’(x) = 2x
'The derivative of f(x) equals 2x'
or simply 'f-dash of x equals 2x'

5 applications of the derivative part 2ap calculus class

Let's try another example.

Example: What is x3 ?

We know f(x) = x3, and can calculate f(x+Δx) :

5 applications of the derivative part 2ap calculus using
Start with:f(x+Δx) = (x+Δx)3
Expand (x + Δx)3:
Put in f(x+Δx) and f(x):x3 + 3x2 Δx + 3x (Δx)2 + (Δx)3 − x3Δx
Simplify (x3 and −x3 cancel):3x2 Δx + 3x (Δx)2 + (Δx)3Δx
Simplify more (divide through by Δx): = 3x2 + 3x Δx + (Δx)2

Result: the derivative of x3 is 3x2

Have a play with it using the Derivative Plotter.

Derivatives of Other Functions

We can use the same method to work out derivatives of other functions (like sine, cosine, logarithms, etc).

5 Applications Of The Derivative Part 2ap Calculus Pdf

But in practice the usual way to find derivatives is to use:

2ap

Example: what is the derivative of sin(x) ?

On Derivative Rules it is listed as being cos(x)

Done.

5 Applications Of The Derivative Part 2ap Calculus Using

Using the rules can be tricky!

5 Applications Of The Derivative Part 2ap Calculus Class

Example: what is the derivative of cos(x)sin(x) ?

You can't just find the derivative of cos(x) and multiply it by the derivative of sin(x) ... you must use the 'Product Rule' as explained on the Derivative Rules page.

It actually works out to be cos2(x) − sin2(x)

5applications Of The Derivative Part 2ap Calculus

So that is your next step: learn how to use the rules.

Notation

'Shrink towards zero' is actually written as a limit like this:


'The derivative of f equals the limit as Δx goes to zero of f(x+Δx) - f(x) over Δx'

Or sometimes the derivative is written like this (explained on Derivatives as dy/dx):

The process of finding a derivative is called 'differentiation'.

Where to Next?

Go and learn how to find derivatives using Derivative Rules, and get plenty of practice: