When y is defined as a function of x

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In summary, the problem asks to find the derivative of y, a function of x, given the equation y + e^y = x^2. The solution involves using implicit differentiation, which results in the derivative of y being equal to 2x divided by 1+e^y.
  • #1
rab-c
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Homework Statement



find [itex]\frac{dy}{dx}[/itex] when y is defined as a function of x by the equation
y+[itex]e^y = x^2[/itex]

Homework Equations





The Attempt at a Solution



hi all,
do i use implicit differentiation for this? I am not really sure how to start...
 
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  • #2
rab-c said:

Homework Statement



find [itex]\frac{dy}{dx}[/itex] when y is defined as a function of x by the equation
y+[itex]e^y = x^2[/itex]

Homework Equations





The Attempt at a Solution



hi all,
do i use implicit differentiation for this? I am not really sure how to start...

Should you use implicit differentiation? Well, why don't you try it, to see what happens?

RGV
 
  • #3


Differentiate both sides of the equation with respect to x. Since the y on the left side is an unknown function of x, yes, you will need to use implicit differentiation.

(But, as Ray Vickson implies, even if you were not sure, you should have tried. Much of mathematics is "try and see if it works".)
 
  • #4


y'+[itex]e^y y' [/itex] = 2x
y'(1+[itex]e^y[/itex]) = 2x
y' = [itex]\frac{2x}{1+e^y}[/itex]
I think this is the right answer?
 

1. What does it mean when y is defined as a function of x?

When y is defined as a function of x, it means that the value of y is dependent on the value of x. In other words, for every input value of x, there will be a corresponding output value of y.

2. How is a function of x different from a regular equation?

A function of x is a type of equation in which the output value (y) is determined by the input value (x), whereas a regular equation may not have this type of relationship between the variables.

3. Can there be more than one function of x for a given value of x?

No, there can only be one function of x for a given value of x. This is because a function must have a unique output value for every input value. However, there can be multiple x values for a given y value in some cases.

4. Are there any restrictions on the values of x and y in a function?

Yes, there are some restrictions on the values of x and y in a function. For example, a function cannot have the same input value (x) for multiple output values (y). Additionally, certain functions may have domain or range restrictions that limit the possible values of x and y.

5. How is a function of x represented graphically?

A function of x is typically represented graphically using a coordinate plane, with the x-axis representing the input value and the y-axis representing the output value. The function can then be plotted as a line or curve on the graph.

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