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### Course: Differential Calculus (2017 edition)>Unit 6

Lesson 3: Polynomial functions differentiation

# Basic derivative rules

Let's explore how to find the derivative of any polynomial using the power rule and additional properties. The derivative of a constant is always 0, and we can pull out a scalar constant when taking the derivative. Furthermore, the derivative of a sum of two functions is simply the sum of their derivatives. Created by Sal Khan.

## Want to join the conversation?

• Surely, at , the rule does also works if n = 0, since the derivative will equal 0.x^-1, which equals 0, and that is the derivative of a constant.
• : It won't work if n=0, then the derivative would be x^(-1) and that is the same thing as 1/(x^1) and if x happened to be equal to zero, the derivative would be 1/0 (one divided by zero) which is undefined.
• Is there a video where the d/dx notation is explicitly explained? It seems like he justs starts using it from here onward without really talking about it.
• d/dx is just like a operator of differentiation. d(y)/dx will mean taking the derivative of y with respect to x.
The d is for delta or difference so basically it means a change in y with a change in x which gives the derivative or the instantaneous slope at a point.
• Why any number raised to zero power becomes 1?
• For all numbers x (not zero) and all numbers m and n ,
x^m
------ = x^(m-n)
x^n
This simply means when you are dividing, and the bases are the same, you SUBTRACT the exponents.

3^1
------ = 3^(1-1) = 3^0 ; but 3/3 = 1 then we conclude that 3^0 = 1
3^1
But isn't only the number 3. All the numbers, that are different from zero, raised to power 0 are equal to one.
• What is a second derivative...is it like an antiderivative
• No, the second derivative is the derivative of the first derivative of any function f(x). It is the change of the rate of change, essentially. The antiderivative, on the other hand, is going backwards from the derivative to the original function. (Later, in calculus, the more common name for the antiderivative becomes the indefinite integral - see integration videos) It is not the same thing as the second derivative.
`d/dx [f(x)*g(x)]`
Does that become:
`[d/dx f(x)][d/dx g(x)]`
• Why we can take the constant out directly for the derivative of a constant times the function?
• The product rule is a little bit more than you need for showing this kind of thing. Suppose you've got a function f(x) (and its derivative) in mind and you want to find the derivative of the function g(x) = 2f(x). By the definition of a derivative this is the limit as h goes to 0 of:
(g(x+h) - g(x))/h = (2f(x+h) - 2f(x))/h = 2(f(x+h) - f(x))/h
Now remember that we can take a constant multiple out of a limit, so this could be thought of as 2 times the limit as h goes to 0 of
(f(x+h) - f(x))/h
Which is just 2 times f'(x) (again, by definition). The principle is known as the linearity of the derivative.
• I don't understand why the derivative of a constant is 0. If f(x) = constant, then slope of f(x) = 0. But then, wouldn't the slope of the tangent(the derivative) be different?
• What's going on here is that the tangent line to any line is just the same line again. Think of it this way, the tangent line at x=a should be the line (out of all possible lines) that approximates f(x) the best near x=a. The line that best approximates f(x) = c is the same line again.
• For the example at the very end of the video, the derivative starts with 6x^2, does that mean that the tangent of the function is a curve? How does that work?
• First, remember that the derivative of a function is the slope of the tangent line to the function at any given point.

If you graph the derivative of the function, it would be a curve. Remember though, that this is not the tangent line to the curve, it is only a graph of the derivative, or the slope of the tangent line to the curve at a given point. You can use this graph to find the derivative at a certain point.

For example, let's look at only the first term in the last example in the video, and its derivative. The term is 2x³, and its derivative is 6x².

The graph of 2x³ will look similar to the graph of x³, an odd function moving from the third quadrant towards the first quadrant. The graph of 6x² will look like the typical graph of a quadratic function, which is some variation of a parabola.

Now, let's try to imagine the tangent line to 2x³ at x = 0. Is the tangent line a parabola? No, that's not a line. So, how can we even find a tangent line from the derivative? Finding the value of the derivative at the x-value, and using that as the tangent line's slope. (After all, the derivative is commonly defined as the slope of the tangent line to the function at that x-value.)

At x = 0, the value of 6x² is 0. Thus, the tangent line is a line with slope 0, or a flat line along y = 0 (the value of x³ evaluated at x = 0).

Furthermore, the derivative is a curve because the slope of the tangent line to the function is changing. Think of this as the function increasing or decreasing faster in some intervals, and not so much in others. At x = 0, the derivative is 0. At x = 0.5, x³ is beginning to increase faster, and the derivative is 1.5. At x = 1, the derivative is 6. At x = 2, the derivative is 24. The derivative is clearly not changing at a constant rate with x.