Question

$\sf Differentiate \: the \:following \:function \:w.r.t \: x$


$\sf log ( {a}^{4x} {(\frac{x-5} {x+4})}^{\frac{3}{4}})$



$\sf The \: Answer \: is \: 4log a + \frac {3}{4 (x-5)} - \frac{3}{4 (x+4)}$



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$\sf Thanks \:Godslayer007\: and\: debismita\: for \:answer$​

Answer 1

Answer--->

dy/dx = 4loga + 3/ { 4 ( x - 5 ) }- 3/ {4 ( x + 4 ) }

Step-by-step explanation:

Given---->

log { a⁴ˣ ( x - 5 / x + 4 )³/⁴ }

To find----> Derivative of given function

Solution----> 1) Plzz see the attachment

2) We know that ,

a) log ( m n ) = logm + logn

b) log ( m/n ) = logm - logn

c) log mⁿ = n logm

3) Applying these properties of log we get,

y = 4x loga + 3/4 log ( x - 5 ) - 3/4 log ( x + 4 )

4) Differentiating with respect to x and applying some formulee of differetiation as follows,

d/dx ( x ) = 1

d/dx ( logx ) = 1/x

we get the answer

Answer 2

Solution

$\sf The \: Answer \: is \: 4log a + \dfrac {3}{4 (x-5)} - \dfrac{3}{4 (x+4)}$

Given

$\sf Let \ y = log ( {a}^{4x} {(\dfrac{x-5} {x+4})}^{\frac{3}{4}})$

To finD

Derivative of y

Now,

$\sf y = log ( {a}^{4x} {(\dfrac{x-5} {x+4})}^{\dfrac{3}{4}}) \\ \\ \longrightarrow \: \sf \: y =( log \: {a}^{4x} ) + \bigg(log \:( \dfrac{x - 5}{x + 4} ){}^{ \frac{3}{4} } \bigg)$

$\longrightarrow \: \sf \:y = \: 4x(log \: a) + \dfrac{3}{4} \{log \: ( \dfrac{x - 5}{x + 4} )\} \\ \\ \longrightarrow \: \sf \: y = 4x(log \: a) + \bigg( \: \dfrac{3}{4} log(x - 5) - \dfrac{3}{4} log(x + 4) \bigg)$

Differentiating y with respect to x,we get :

$\longrightarrow \: \sf \: y' = \dfrac{d(4 x\: log \: a )}{dx} + \dfrac{3}{4} \times \dfrac{d( log( x - 5)) }{dx} - \dfrac{3}{4} \times \dfrac{d( log(x + 4)) }{dx} \\ \\ \longrightarrow \: \sf \: y' = \bigg( 4 log(a) \times \dfrac{dx}{dx} \bigg)+ \bigg(\dfrac{3}{4} \times \dfrac{1}{x - 5} \bigg) - \bigg( \dfrac{3}{4} \times \dfrac{1}{x + 4} \bigg) \\ \\ \longrightarrow \boxed{ \boxed{ \sf \: y' = 4 log(a) + \dfrac{3}{4(x + 5)} - \dfrac{3}{4(x - 4)} }}$

Note

• log(ab) = log(a) + log(b)

• log(a/b) = log(a) - log(b)

• log(aⁿ) = n log(a)

• d(log x)/dx = 1/x