Through the implicit function theorem, we are guaranteed that y can be represented as a single variable function of x, this helps a lot with finding that limit. Implicit differentiation is a way of differentiating when you have a function in terms of both x and y. For example: #x^2+y^2=16# This is the formula for a circle with a centre at (0,0) and a radius of 4. 6.3 Explicit Vs Implicit Differentiation Notes No tes Key. Homework Hw Key. Powered by Create your own unique website with customizable templates.
What is the difference between implicit and explicit differentiation?
1 Answer
It is a difference in how the function is presented before differentiating (or how the functions are presented).
Explanation:
#y = -3/5x+7/5# gives #y# explicitly as a function of #x#.
#3x+5y=7# gives exactly the same relationship between #x# and #y#, but the function is implicit (hidden) in the equation. To make the function explicit, we solve for #x#
In #x^2+y^2=25#, #y# is not a function of #x#. However, there are two functions implicit in the equation. We can make the functions explicit by solving for #y#.
#y = +- sqrt(25-x^2)# is equivalent to the equation above and it has 2 functions that are not too difficult to make explicit:
Hotel dash free. download full version for mac trial video editing software. #y=sqrt(25-x^2)# gives #y# as a function of #x# and
#y=-sqrt(25-x^2)# gives #y# as a different function of #x#.
We can differentiate either the implicit or explicit presentations.
Differentiating implicitly (leaving the functions implicit) we get
PEGI. #2x+2y dy/dx = 0##' '# so #' '##dy/dx = -x/y#
The #y# in the formula for the derivative is the price we pay for not making the function explicit. It replaces the explicit form of the function, whatever that may be.
For #y=sqrt(25-x^2)#, we get #dy/dx = - x/sqrt(25-x^2)# (use the power and chain rule), and
6.1 Implicit Vs Explicitap Calculus Calculator
for #y= - sqrt(25-x^2)#, we get #dy/dx = x/sqrt(25-x^2)#.
The equation #y^5+4x^2y^2-3y+7x=28 # cannot be solved algebraically for #y#, (or anyway, some 5th degree equations cannot be solved) but there are several functions of #x# implicit in the equation. You can see them in the graph of the equation (shown below).
graph{y^5+4x^2y^2-3y+7x=28 [-7.14, 6.91, -4.66, 2.36]}
6.1 Implicit Vs Explicitap Calculus 2nd Edition
We can cut the graph into pieces, each of which is the graph of some function of #x# on some domain.
Implicit differentiation allow us to find the derivative(s) of #y# with respect to #x#without making the function(s) explicit. Doing that, we can find the slope of the line tangent to the graph at the point #(1,2)#.
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