Question

can somone explain IMPLICIT differentiation and why it is used

Answer 1

sure it is used when you have an equation involving x and y but you do not or cannot solve for y

Answer 2

To make your life difficult

Answer 3

LOL @ IMRANMEAH91

Answer 4

this was in fact the original problem. not "find the derivative of this or that" but you have a curve defined by something like
\[x^2+xy+y^2=10\] and you want to find the slope of the tangent line at a given point

Answer 5

the function stuff came later

Answer 6

so can you show me implicit differentitiation on that question

Answer 7

sure. but just to repeat myself the idea of functions came later. curves were given by equations involving two variables, not y = something. ok here we go

Answer 8

we want
\[\frac{dy}{dx}=y'\]

Answer 9

My question is, why would anyone want to find a point on a curve? I'd rather do more interesting things in life.

Answer 10

starting with
\[x^2+xy+y^2=10\] we get
\[2x+xy'+y+2yy'=0\] and then use algebra to solve for
\[y'\]

Answer 11

steps were
1) derivative of
\[x^2\] is \[2x\]
2) the derivative of
\[xy\] requires product rule because it is a product. the derivative of
\[x\] is
\[1\] and the derivative of
\[y \] is
\[y'\]

Answer 12

where did 2yy' come from?

Answer 13

ok lets go slow

Answer 14

we are viewing y as a function of x, even though we did not solve for y

Answer 15

in other words we are thinking
\[y=f(x)\] we just don't know it

Answer 16

so we have
\[y^2=f^2(x)\]

Answer 17

and we take the derivative using the chain rule, which is why these problems come right after the chain rule problems in the calc text

Answer 18

by the chain rule, the derivative of
\[f^2(x)\] is
\[2f(x)f'(x)\]

Answer 19

here we have
\[f(x)=y,f'(x)=y'\] giving hte derivative of
\[y^2\] as \[2yy'\]

Answer 20

just like the derivative of
\[\sin^2(x)=2\sin(x)\cos(x)\] but more general

Answer 21

oh ok