When they want you to take the derivative of an equation the directions can say many different things that mean they want a derivative. y¹, f¹(x), dy/dx, d/dx, dx[y], slope of a tangent line, and the slope of the curve at a point is all the different ways they can ask for a derivative. When the directions say to use the definition of a derivative to find the derivative it means to use the formula f(x + Δx)-f(x)/Δx, do not use the short cut way. Here is some examples of some directions that they want you to take the derivative of something: EX: Find dy/dx of x²+1 at the points (0, 1) and (-1, 2). EX: Find f¹(x) for f(x)=√x. then find the slopes of the graph of f at the points (1, 1) and (4, 2). Discuss the behavior of f at (0, 0). EX: Find the slope of f(x)= 2x-3 at (2, 1) EX: Dx [3x²+4x+2] at x=-1
There are quite a few different ways for the directions to tell you how to solve for a derivative. Those ways are: y' f'(x) dy/dx d/dx Dx[y] slope of a tangent line slope of the curve at a point
Examples of what the directions might say: 1) Find dy/dx of x^3 at the poing (0,1) & (-1,2) 2) Find d/dx [x^3+2x] 3) Find f'(4) where f(x)=5x^2-8x 4) Find y' x^3 5) Find the slope of a tangent line of f(x)=x^2 6) Find the slope of the curve at a point where y'= x^2+1 at points (0,1) & (-1,2)
When a problem is askin for you to take a derivative it has a few different notations. These notations can include: use the derivative formula, find y^1 ( y prime ), find f^1 ( x ), find dy / dx, find dx ( y ). It could also ask you for the slope of a tangent line, or even the slope of a curve at a certain point. It is important to know all of these because any of them can be used when askin you to find a derivative. Examples of this: 1 ) Find dy / dx of the equation x^3 + 2x^2 – 4 2 ) Find the slope of the tangent line x^2 + 5 3 ) What is ( f prime of x ) of the x^2 – 3x at the point ( 3, 2 ) 4 ) Find the slope of x^2 + 4 at the point ( 4, 7 )
There are several different ways for a problem to ask for a derivative. These are some of the things you should look for in a problem in order to find out whether or not it is asking for the derivative...
dy/dx d/dx Dx[y] y^1 f^1(x) slope of a tangent line ...or if a problem asks you to use the formula for a derivative.
Here are some examples of directions that ask for the derivative...
Ex#1. Find k such that the line is tangent to the graph of the function.
1.) f(x)=x^2-k(x) line:y=5x-4
Ex#2. Find a and b such that f is differentiable everywhere.
When taking a derivative you are asked in a few different ways. The following ways are: y' f'(x) dy/dx d/dx Dx[y] slope of a tangent line slope of the curve at a point You may also have to use the derivative formula which is (x+deltax)-f(x)/deltax
All asking you to find a derivative...
ReplyDeleteFind:
dy/dx
y^1
f^1(x)
d/dx
Dx[y]
slope of a tangent line
slope of the curve at a point
Find dy/dx of x^2+1 at the points
(0,1) and (-1,2)
lim
Δx->0 (x+Δx)^2 + 1 -(x^2 + 1)/Δx
(x+Δx)(x+Δx)
x^2+2xΔx+Δx^2+1-x^2-1/Δx
*cancel and factor a Δx out.
Δx(2x+Δx)/Δx
=2x
slope @ (0,1) = 2(0)=0
slope @ (-1,2) = 2(-1) = -2
When they want you to take the derivative of an equation the directions can say many different things that mean they want a derivative. y¹, f¹(x), dy/dx, d/dx, dx[y], slope of a tangent line, and the slope of the curve at a point is all the different ways they can ask for a derivative. When the directions say to use the definition of a derivative to find the derivative it means to use the formula f(x + Δx)-f(x)/Δx, do not use the short cut way. Here is some examples of some directions that they want you to take the derivative of something:
ReplyDeleteEX: Find dy/dx of x²+1 at the points (0, 1) and (-1, 2).
EX: Find f¹(x) for f(x)=√x. then find the slopes of the graph of f at the points (1, 1) and (4, 2). Discuss the behavior of f at (0, 0).
EX: Find the slope of f(x)= 2x-3 at (2, 1)
EX: Dx [3x²+4x+2] at x=-1
There are many keywords to look for when finding dervatives
ReplyDeletesuch as y^1, f^1(x), dy/dx, d/dx, dx[y]
slope of a tan line, slope of the curve at a point
here is an example
f(x)= 2x^2+1
find f^1(x)
lim
deltax-->0
you would plug in the formula
2(x+deltax)^2+1-(2x^2+1)
2(x^2+2xdeltax+deltax^2)+1-2x^2-1
2x^2+4xdeltax+2deltax^2+1-2x^2-1
4xdeltax+2deltax^2/deltax
deltax(4x+2deltax)/deltax
the final answer is 4x because that is what you
get when you plug it in.
There are quite a few different ways for the directions to tell you how to solve for a derivative. Those ways are:
ReplyDeletey'
f'(x)
dy/dx
d/dx
Dx[y]
slope of a tangent line
slope of the curve at a point
Examples of what the directions might say:
1) Find dy/dx of x^3 at the poing (0,1) & (-1,2)
2) Find d/dx [x^3+2x]
3) Find f'(4) where f(x)=5x^2-8x
4) Find y' x^3
5) Find the slope of a tangent line of f(x)=x^2
6) Find the slope of the curve at a point where
y'= x^2+1 at points (0,1) & (-1,2)
When a problem is askin for you to take a derivative it has a few different notations. These notations can include: use the derivative formula, find y^1 ( y prime ), find f^1 ( x ), find dy / dx, find dx ( y ). It could also ask you for the slope of a tangent line, or even the slope of a curve at a certain point. It is important to know all of these because any of them can be used when askin you to find a derivative.
ReplyDeleteExamples of this:
1 ) Find dy / dx of the equation x^3 + 2x^2 – 4
2 ) Find the slope of the tangent line x^2 + 5
3 ) What is ( f prime of x ) of the x^2 – 3x at the point ( 3, 2 )
4 ) Find the slope of x^2 + 4 at the point ( 4, 7 )
There are several different ways for a problem to ask for a derivative. These are some of the things you should look for in a problem in order to find out whether or not it is asking for the derivative...
ReplyDeletedy/dx
d/dx
Dx[y]
y^1
f^1(x)
slope of a tangent line
...or if a problem asks you to use the formula for a derivative.
Here are some examples of directions that ask for the derivative...
Ex#1. Find k such that the line is tangent to the graph of the function.
1.) f(x)=x^2-k(x) line:y=5x-4
Ex#2. Find a and b such that f is differentiable everywhere.
120.) Prove that d/dx[cosx]=-sinx
When taking a derivative you are asked in a few different ways. The following ways are:
ReplyDeletey'
f'(x)
dy/dx
d/dx
Dx[y]
slope of a tangent line
slope of the curve at a point
You may also have to use the derivative formula which is (x+deltax)-f(x)/deltax