High School Calculus/Implicit Differentiation

Implicit Differentiation

When a functional relation between x and y cannot be readily solved for y, the preceding rules may be applied directly to the implicit function.

The derivative will usually contain both x and y. Thus the derivative of an algebraic function, defined by setting the polynomial of x and y to zero.

Ex. 1

Given the function y of x

$x^5+y^5-5xy+1=0$

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$

Since

$\frac{\mathrm{d}}{\mathrm{d}x}(x^5+y^5-5xy+1)=0$

$=5x^4+5y^4\frac{\mathrm{d}y}{\mathrm{d}x}-5y-5x\frac{\mathrm{d}y}{\mathrm{d}x}=0$

In solving for $\frac{\mathrm{d}y}{\mathrm{d}x}$ we must first factor the differentiation problem

In doing this we get

$\frac{\mathrm{d}y}{\mathrm{d}x}(5y^4-5x)+(5x^4-5y)=0$

From here we subtract the $\frac{\mathrm{d}y}{\mathrm{d}x}$ to one side

Thus giving us

$5x^4-5y=-\frac{\mathrm{d}y}{\mathrm{d}x}(-5x+5y^4)$

Here I am going to skip a step in solving this implicit differentiation problem. I am going to skip the step where I divide the -1 over to the other side.

From here we divide the polynomial from the $\frac{\mathrm{d}y}{\mathrm{d}x}$ over to the other side. Giving us

$\left(\frac{-5x^4+5y}{-5x+5y^4}\right)=\frac{\mathrm{d}y}{\mathrm{d}x}$

Now we simplify and get

$\frac{\mathrm{d}y}{\mathrm{d}x}=\left(\frac{x^4-y}{x-y^4}\right)$

Other problems to work on

Ex. 2

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ given the function

$x{y}^2+x^{2}y=1$

Ex. 3

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ given the function

$x+y+(x-y{)}^2+(2x-3y{)}^3=0$