Implicit differentiation calculator

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Solution (Print / Save)

Solution provided by AtoZmath.com

1. `x^2+y^2=15`, derivative with respect to x

2. `x^3+y^3=2xy`, derivative with respect to x

3. `x^2+xy+y^2=9`, derivative with respect to x

Example (with steps)

`f(x,y)=x^2+y^2=15`
Find Implicit differentiation, Derivative with respect to x

Solution:
Method-1: using Chain Rule
`x^2+y^2-15=0`

After implicit differentiation

`d/(dx)(x^2+y^2-15)=2x+2yy'`

`d/(dx)(x^2+y^2-15)`

`=d/(dx)(x^2)+d/(dx)(y^2)-d/(dx)(15)`

`d/(dx)(y^2)=2yy'`

`d/(dx)(y^2)`

`=2y*d/(dx)(y)`

`=2y*y'`

`=2yy'`

`=2x+2yy'-0`

`=2x+2yy'`

`2x+2yy'=0`

`=>2yy'+2x = 0`

`=>2yy' = -2x`

`=>y' = (-2x)/(2y)`

`=>y' = (-x)/y`

`=>y' = -x/y`

Method-2: using partial derivatives `Fx,Fy`. If `F(x,y)=0`, then `y'=-(Fx)/(Fy)`

`Fx=(del)/(del x)(x^2+y^2-15)=2x`

Now,
`del/(delx)(x^2+y^2-15)`

`=del/(delx)(x^2)+del/(delx)(y^2)-del/(delx)(15)`

`del/(delx)(y^2)=0`

`del/(delx)(y^2)`

`=2y*del/(delx)(y)`

`=2y*0`

`=0`

`=2x+0-0`

`=2x`

`:.(del)/(del x)(x^2+y^2-15)=2x`

`Fy=(del)/(del y)(x^2+y^2-15)=2y`

Now,
`del/(dely)(x^2+y^2-15)`

`=del/(dely)(x^2)+del/(dely)(y^2)-del/(dely)(15)`

`del/(dely)(x^2)=0`

`del/(dely)(x^2)`

`=2x*del/(dely)(x)`

`=2x*0`

`=0`

`=0+2y-0`

`=2y`

`:.(del)/(del y)(x^2+y^2-15)=2y`

`y'=-(Fx)/(Fy)`

`:.y'=-(2x)/(2y)`

`:.y'=(-x)/y`

Input functions

Sr No.	Function	Input value
1.	`x^3`	x^3
2.	`sqrt(x)`	sqrt(x)
3.	`root(3)(x)`	root(3,x)
4.	sin(x)	sin(x)
5.	cos(x)	cos(x)
6.	tan(x)	tan(x)
7.	sec(x)	sec(x)
8.	cosec(x)	csc(x)
9.	cot(x)	cot(x)
10.	`sin^(-1)(x)`	asin(x)
11.	`cos^(-1)(x)`	acos(x)
12.	`tan^(-1)(x)`	atan(x)
13.	`sin^2(x)`	sin^2(x)
14.	`log_y(x)`	log(y,x)
15.	`log_10(x)`	log(x)
16.	`log_e(x)`	ln(x)
17.	`e^x`	exp(x) or e^x
18.	`e^(2x)`	exp(2x) or e^(2x)
19.	`oo`	inf