Question

Differentiate :
${a}^{x}( x + {1 \div x})^{10}$
​

Answer 1

GIVEN :–

• A function $\bf \: {a}^{x} \bigg( x + \dfrac{1}{ x} \bigg)^{10}$

TO FIND :–

• Differentiate form = ?

SOLUTION :–

• Let the function be –

$\\ \bf \implies y = {a}^{x} \bigg( x + \dfrac{1}{ x} \bigg)^{10}$

• Using identity –

$\\ \bf \implies \dfrac{d(u.v)}{dx} = u \dfrac{dv}{dx} + v \dfrac{du}{dx}$

• So that –

$\\ \bf \implies \dfrac{dy}{dx} = {a}^{x}. \dfrac{d}{dx} \bigg( x + \dfrac{1}{ x} \bigg)^{10} + \bigg( x + \dfrac{1}{ x} \bigg)^{10}. \dfrac{d( {a}^{x})}{dx}$

$\\ \bf \implies \dfrac{dy}{dx} = {a}^{x}. (10) \bigg( x + \dfrac{1}{ x} \bigg)^{10 - 1} \bigg(1 - \dfrac{1}{ {x}^{2} } \bigg)+ \bigg( x + \dfrac{1}{ x} \bigg)^{10}. {a}^{x} log(x)$

$\\ \bf \implies \dfrac{dy}{dx} = 10{a}^{x}\bigg( x + \dfrac{1}{ x} \bigg)^{9} \bigg(1 - \dfrac{1}{ {x}^{2} } \bigg)+ \bigg( x + \dfrac{1}{ x} \bigg)^{10}. {a}^{x} log(x)$

$\\ \bf \implies \dfrac{dy}{dx} = 10{a}^{x}\bigg( x + \dfrac{1}{ x} \bigg)^{9} \bigg(1 - \dfrac{1}{ {x}^{} } \bigg)\bigg(1 + \dfrac{1}{ {x}^{} } \bigg)+ \bigg( x + \dfrac{1}{ x} \bigg)^{10}. {a}^{x} log(x)$

$\\ \bf \implies \dfrac{dy}{dx} = 10{a}^{x}\bigg( x + \dfrac{1}{ x} \bigg)^{10} \bigg(1 - \dfrac{1}{ {x}^{} } \bigg)+ \bigg( x + \dfrac{1}{ x} \bigg)^{10}. {a}^{x} log(x)$

$\\ \bf \implies \dfrac{dy}{dx} ={a}^{x}\bigg( x + \dfrac{1}{ x} \bigg)^{10} \bigg \{10 \bigg(1 - \dfrac{1}{ {x}^{} } \bigg)+ log(x) \bigg \}$

$\\ \bf \implies \dfrac{dy}{dx} ={a}^{x}\bigg( x + \dfrac{1}{ x} \bigg)^{10} \bigg \{10 - \dfrac{10}{x} + log(x) \bigg \}$