Respuesta :

leena

Hi there!

[tex]f(x) = x^{cosx}[/tex]

We can rewrite as:
[tex]y = x^{cosx}[/tex]

Take the natural log of both sides.

[tex]lny = cos(x)ln(x)[/tex]

Now, differentiate both sides. Recall the product rule:


[tex]\frac{d}{dx}f(x) * g(x) = f'(x)g(x) + g'(x)f(x)[/tex]

[tex]f(x) = cos(x)\\\\g(x) = ln(x)[/tex]

And the natural log rule:
[tex]\frac{dy}{dx} (lnx) = \frac{1}{x}[/tex]

Using these rules:

[tex]\frac{1}{y}\frac{dy}{dx} = cos(x)(\frac{1}{x}) -sin(x)ln(x)[/tex]

Multiply both sides by y.

[tex]\frac{dy}{dx} = y(cos(x)(\frac{1}{x}) -sin(x)ln(x))[/tex]

Since we cannot have a 'y' in the equation, we can substitute in the original expression above:
[tex]y = x^{cosx}\\\\[/tex]

Thus:
[tex]\frac{dy}{dx} = x^{cosx}(cos(x)(\frac{1}{x}) -sin(x)ln(x))[/tex]

[tex]\boxed{\frac{dy}{dx} = x^{cosx}(\frac{cos(x)}{x} -sin(x)ln(x))}[/tex]

We are given with a function and have to find it's Derivative , So let's start !!

[tex]{:\implies \quad \sf f(x)=x^{\cos (x)}}[/tex]

Take Natural log on both sides ;

[tex]{:\implies \quad \sf ln\{f(x)\}=ln\{x^{\cos (x)}\}}[/tex]

[tex]{:\implies \quad \sf ln\{f(x)\}=\cos (x)ln(x)\quad \qquad \{\because ln(a^{b})=b\: ln(a)\}}[/tex]

Differentiating both sides w.r.t.x ;

[tex]{:\implies \quad \sf \dfrac{1}{f(x)}\cdot f^{\prime}(x)=\dfrac{d}{dx}\bigg\{\cos (x)\cdot ln(x)\bigg\}}[/tex]

Using Leibnitz's Product rule ;

[tex]{:\implies \quad \sf f^{\prime}(x)=f(x)\left[\dfrac{d}{dx}\{\cos (x)\}\cdot ln(x)+\dfrac{d}{dx}\{ln(x)\}\cdot \cos (x)\right]}[/tex]

[tex]{:\implies \quad \sf f^{\prime}(x)=x^{\cos (x)}\bigg\{- \sin (x)ln(x)+\dfrac{\cos (x)}{x}\bigg\}}[/tex]

[tex]{:\implies \quad \bf \therefore \quad \underline{\underline{f^{\prime}(x)=x^{\cos (x)}\bigg\{\dfrac{\cos (x)}{x}- \sin (x)ln(x)\bigg\}}}}[/tex]

Used Concepts :-

  • [tex]{\bf \dfrac{d}{dx}\{ln(x)\}=\dfrac1x}[/tex]

  • [tex]{\bf \dfrac{d}{dx}\{\cos (x)\}=- \sin (x)}[/tex]

Leibnitz's Product Rule of differentiation :-

  • [tex]{\quad \boxed{\bf{\dfrac{d}{dx}(uv)=v\dfrac{du}{dx}+u\dfrac{dv}{dx}}}}[/tex]

Where , u & v both are functions of x .