Math, asked by l4731306, 8 months ago

differentiate the following wrt x 1/x^3/2​

Answers

Answered by rishiramanuja
1

Answer:

Let y=x  

x  

2

−3

+(x−3)  

x  

2

 

 

Also, let u=x  

x  

2

−3

 and v=(x−3)  

x  

2

 

 

∴y=u+v

Differentiating both sides with respect to x, we obtain  

dx

dy

​  

=  

dx

du

​  

+  

dx

dv

​  

 .....(1)

u=x  

x  

2

−3

 

∴logu=log(x  

x  

2

−3

)

⇒logu=(x  

2

−3)logx

Differentiating with respect to x, we obtain  

u

1

​  

 

dx

du

​  

=logx.  

dx

du

​  

=logx.  

dx

d

​  

(x  

2

−3)+(x  

2

−3).  

dx

d

​  

(logx)  

⇒  

u

1

​  

 

dx

du

​  

=logx.2x+(x  

2

−3).  

x

1

​  

 

dx

du

​  

=x  

x  

2

−3

[  

x

x  

2

−3

​  

+2xlogx]  

Also,  

v=(x−3)  

x  

2

 

 

∴logv=log(x−3)  

x  

2

 

 

⇒logv=x  

2

log(x−3)

Differentiating both sides with respect to x, we obtain  

v

1

​  

 

dx

dv

​  

=log(x−3).  

dx

d

​  

(x  

2

)+x  

2

.  

dx

d

​  

[log(x−3)]  

⇒  

v

1

​  

 

dx

dv

​  

=log(x−3).2x+x  

2

.  

x−3

1

​  

.  

dx

d

​  

(x−3)  

⇒  

dx

dv

​  

=v[2xlog(x−3)+  

x−3

x  

2

 

​  

.1]  

⇒  

dx

dv

​  

=(x−3)  

x  

2

 

[  

x−3

x  

2

 

​  

+2xlog(x−3)]

Substituting the expressions of  

dx

du

​  

 and  

dx

dv

​  

 in equation (1), we obtain

dx

dy

​  

=x  

x  

2

−3

[  

x

x  

2

−3

​  

+2xlogx]+(x−3)  

x  

2

 

[  

x−3

x  

2

 

​  

+2xlog(x−3)]

Step-by-step explanation:

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