Question

y^3*x^5=(x+y)^8 then dy/dx is

Answer 1

Question:

If (y^3)(x^5) = (x+y)^8 , then find dy/dx .

Answer:

dy/dx = [5•(y^3)•(x^4) - {8(x+y)^7}] /

[{8(x+y)^7} - 3•(x^5)•(y^2)]

Note:

• d(u±v)/dx = du/dx ± dv/dx

• d(u•v)/dx = u•(dv/dx) + v•(du/dx)

• d(x^n)/dx = n•x^(n-1)

Solution:

Here,

The given equation is ;

(y^3)(x^5) = (x+y)^8

Now,

Differentiating both sides of the given equation with respect to x , we get;

=> d{(y^3)(x^5)}/dx = d{(x+y)^8}/dx

=> (y^3)•d(x^5)/dx + (x^5)•d(y^3)/dx

= {8(x+y)^7}•d(x+y)/dx

=> (y^3)•(5x^4) + (x^5)•(3y^2)•(dy/dx)

= {8(x+y)^7}•(dx/dx + dy/dx)

=> 5•(y^3)•(x^4) + 3•(x^5)•(y^2)•(dy/dx)

= {8(x+y)^7}•(1 + dy/dx)

=> 5•(y^3)•(x^4) + 3•(x^5)•(y^2)•(dy/dx)

= {8(x+y)^7} + {8(x+y)^7}•(dy/dx)

=> 5•(y^3)•(x^4) - {8(x+y)^7} =

{8(x+y)^7}•(dy/dx) - 3•(x^5)•(y^2)•(dy/dx)

=> 5•(y^3)•(x^4) - {8(x+y)^7} =

(dy/dx)•[{8(x+y)^7} - 3•(x^5)•(y^2)]

=> dy/dx = [5•(y^3)•(x^4) - {8(x+y)^7}] /

[{8(x+y)^7} - 3•(x^5)•(y^2)]

Hence,

The required value of dy/dx is;

[5•(y^3)•(x^4) - {8(x+y)^7}] / [{8(x+y)^7} - 3•(x^5)•(y^2)]

Answer 2

Question :-

$\small \: \: if \: \: {y}^{3} {x}^{5} = {(x + y)}^{8} \: \: then \: find \: \: \frac{dy}{dx}$

Answer:-

$\boxed{ \red{\small \frac{dy}{dx} =} \green{ \frac{8 { \:(x + y)}^{7} - 5 {x}^{4} {y}^{3} }{ 3 {y}^{2} {x}^{5} - 8 {(x + y)}^{7} } } }\:$

Explanation:-

Applied Conditions:-

We know that,

$\star \: \: \: \frac{d}{dx} {x}^{n} = n \: \: {x}^{n - 1} \\ \\ \star \: \: for \: two \: variables \\ \frac{d}{dx} \: x \: y \: = x \: \frac{dy}{dx} + y \: \frac{d}{dx} x$

To find :-

$\red{find \: \: \frac{dy}{dx} \: } \:$

Solution :-

According to the question,

$\small \: \: {y}^{3} \: {x}^{5} = {(x + y)}^{8}$

Now , differentiating on both sides with respect to "x" ,then we get,

$\small \: \frac{d}{dx} {y}^{3} {x}^{5} = \frac{d}{dx} {(x +y) }^{8}$

Applied the given conditions,

$\small \: {y}^{3} \frac{d}{dx} {x}^{5} + {x}^{5} \frac{d}{dx} {y}^{3} \: = \frac{d}{dx} {(x + y)}^{8} \\ \\ \small \: {y}^{3} \times 5 {x}^{4} + {x}^{5} \times 3 {y}^{2} \frac{dy}{dx} = 8 {(x + y)}^{7} \big(1 + \frac{dy}{dx} \big) \\ \\ \small \: 5 {x}^{4} {y}^{3} + 3 {y}^{2} {x}^{5} \frac{dy}{dx} = 8 {(x + y)}^{7} + 8 {(x + y)}^{7} \frac{dy}{dx} \\ \\ \small \: 3 {y}^{2} {x}^{5} \frac{dy}{dx} - 8 {(x + y)}^{7} \frac{dy}{dx} = 8 {(x + y)}^{7} - 5 {x}^{4} {y}^{3} \\ \\ \small \: \frac{dy}{dx} \bigg(3 {y}^{2} {x}^{5} - 8 {(x + y)}^{7} \bigg) = 8 {(x + y)}^{7} - 5 {x}^{4} {y}^{3} \:$

Now ,

$\red{ \boxed{ \small \frac{dy}{dx} = \frac{8 { \:(x + y)}^{7} - 5 {x}^{4} {y}^{3} }{ 3 {y}^{2} {x}^{5} - 8 {(x + y)}^{7} } }}$

This is the required solution.