Answer to Question #113939 in Calculus for sandeep kumar

Question

Differentiate y w.r.t x
1. ) y =√(x+√(x+√x))
2.y=e^cosx+cos(e^x )
3.y=ln(x lnx)

Answer 1

By using the properties of differentiation of the function of power " n ", where n is constant, we can answer the following question

$\begin{aligned} &\text { 1)} y=\sqrt{(x+\sqrt{x+\sqrt{x}})}\\ &y^{\prime}=\frac{1}{2 \sqrt{(x+\sqrt{x+\sqrt{x}})}} *\left(1+\frac{1+\frac{1}{2 \sqrt{x}}}{2 \sqrt{x+\sqrt{x}}}\right) \end{aligned}$

By using the properties of differentiation of the exponential functions, we get the answer of the following question

$\begin{aligned} &2)\,y=e^{\cos x}+\cos e^{x}\\ &y^{\prime}=(-\sin x) * e^{\cos x}-e^{x}\left(\sin e^{x}\right) \end{aligned}$

Using the properties of differentiation of the Logarithmic functions, we get the answer of the following question

$\begin{aligned} &3)y=\ln (x \ln x)\\ &y^{\prime}=\frac{\ln x+x * \frac{1}{x}}{x \ln x}=\frac{\ln x+1}{x \ln x} \end{aligned}$