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
Find ![{\displaystyle {\operatorname {d} y \over \operatorname {d} x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/402f44e7dfeb414dcdf97000bd6808ad5fe2359a)
Since
![{\displaystyle {\operatorname {d} \over \operatorname {d} x}(x^{5}+y^{5}-5xy+1)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33b32a4c21e5df7916ba6414ad19a6c12f7980b7)
![{\displaystyle =5x^{4}+5y^{4}{\operatorname {d} y \over \operatorname {d} x}-5y-5x{\operatorname {d} y \over \operatorname {d} x}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10141c3181fda6f0de76f0e766a52a43b5ab576a)
In solving for
we must first factor the differentiation problem
In doing this we get
![{\displaystyle {\operatorname {d} y \over \operatorname {d} x}(5y^{4}-5x)+(5x^{4}-5y)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e36f9e4eb273a31208065b2846156d234cf05706)
From here we subtract the
to one side
Thus giving us
![{\displaystyle 5x^{4}-5y=-{\operatorname {d} y \over \operatorname {d} x}(-5x+5y^{4})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93ef0b65046c0f38e41b016214a6c463ec631a3d)
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
over to the other side. Giving us
![{\displaystyle \left({\frac {-5x^{4}+5y}{-5x+5y^{4}}}\right)={\operatorname {d} y \over \operatorname {d} x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f42124fe58a3509668d80ca12aed64366d943e3)
Now we simplify and get
![{\displaystyle {\operatorname {d} y \over \operatorname {d} x}=\left({\frac {x^{4}-y}{x-y^{4}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/262485bc522906255396c6509c57709f9d13cf51)
Other problems to work on
Ex. 2
Find
given the function
![{\displaystyle xy^{2}+x^{2}y=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4683b14bcbde9fab9b1293e532aa20921b1cb452)
Ex. 3
Find
given the function
![{\displaystyle x+y+(x-y)^{2}+(2x-3y)^{3}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf8feb6f8e6d661ef651904bdb7cd858ee72d590)