Derivatives Questions

Question 1

If, in a polynomial, a constant proceeds x^n, how do you derive it, and provide an example.

Answer 1

Seperate the two and find the derivative of x^n; ie 2x^3 is 2 * 3x^2

Question 2

How do mathematical expressions translate from polynomials to the derivative?

Answer 2

They remain the same

Question 3

What does the f’(x) of an f(x) tell you?

Answer 3

Tells you the slope or instantaneous rate of change at every x value of the original function.

Question 4

What if the equation are two factors multiplied and what is an example?

Answer 4

Then you utilise the product rule; if you have Y = (X^2 +1) (X^3 + X + 5) the answer would be; dy/dx = (x^2+1) (3x^2+1) + (x^3+x+5)(2x)

Question 5

What is the difference between dy/dx and d/dx?

Answer 5

dy/dx says you are taking the derivative of y with respect to x, whilst d/dx is taking to derivative of SOMETHING with respect to x

Question 6

What is an example of dy/dx and d/dx with respect to a given y?

Answer 6

The derivative of ( y= x^2 + x )is y’ = 2x + 1, this equation being dy/dx whilst d/dx would look like; (x^2 + x) = 2x+1, explaining what has happened.

Question 7

How does a function f with a particular x value correspond to graph form, assuming no values are given

Answer 7

It is written as (x, f(x)) on the graph

Question 8

What if a graph of f has a gradient at a point (x, f(x))?

Answer 8

Then f is considered differentiable

Question 9

What is an example of f being differentiable?

Answer 9

The function f(x) = x^2 is differentiable at EVERY value of x, however if the graph of f doesnt have a gradient at the point with (x, f(x)), due to no viable tangent at that point, f is NOT differentiable at that point, o f(x) =[x] is not differentiable at 0.

Question 10

Why is thinking of gradients as defining a new function related to f useful?

Answer 10

Because the gradient of graph of f varies depending on value of x you are considering, this function being the derivative of f; (f’(, the domain of the derivative consisting of ALL the values at which f is differentiable

Question 11

What does the value of the derivative of a function at a particular input value x give you?

Answer 11

The derivative of f (x); ie the derivative of the function f(x) = x^2 at x=1 is f’(1) = 2.