# Find the derivative of the following function using logarithmic differentiation?

Mar 30, 2017

see below

#### Explanation:

Steps in Logarithmic Differentiation
1. Take natural logarithms of both sides of an equation y=f(x) and use the Laws of Logarithms to simplify.
2. Differentiate implicitly with respect to x
3. Solve the resulting equation for y’
4. Replace y with original equation

Let $y = \frac{5 \sqrt{x} {\left({x}^{2} + 1\right)}^{4}}{{\left(2 x + 1\right)}^{3} {\tan}^{5} \left({x}^{2}\right)}$ and lets rewrite the equation as $y = \frac{5 {x}^{\frac{1}{2}} {\left({x}^{2} + 1\right)}^{4}}{{\left(2 x + 1\right)}^{3} {\tan}^{5} \left({x}^{2}\right)}$

Now use the following Properties of Logarithms to expand the original problem

color(red)(log_b(xy)=log_b x+log_b y
color(red)(log_b(x/y)=log_b x-log_b y
color(red)(log_bx^n=n log_b x
That is,

color(blue)(ln y=ln((5x^(1/2)(x^2+1)^4)/((2x+1)^3tan^5(x^2)))

color(blue)(ln y =5ln x^(1/2)+ln(x^2+1)^4-ln(2x+1)^3-ln (tan x^2)^5

color(blue)(ln y =5/2 ln x+4 ln(x^2+1)-3ln(2x+1)-5 ln(tan x^2)

color(blue)(d/dx (ln y =5/2 ln x+4 ln(x^2+1)-3ln(2x+1)-5 ln(tan x^2))

color(blue)(1/y dy/dx = 5/2 1/x + 4 ((2x)/(x^2+1))-3(2/(2x+1))-5((2xsec^2(x^2))/tan x^2)

$\textcolor{b l u e}{\frac{1}{y} \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{5}{10 x} + \frac{8 x}{{x}^{2} + 1} - \frac{6}{2 x + 1} - \frac{10 x {\sec}^{2} \left({x}^{2}\right)}{\tan} {x}^{2}}$

color(blue)( dy/dx = [5/(10x)+(8x)/(x^2+1)-6/(2x+1)-(10xsec^2(x^2))/tan x^2]*y

color(blue)( dy/dx = [5/(10x)+(8x)/(x^2+1)-6/(2x+1)-(10xsec^2(x^2))/tan x^2]*[(5sqrtx (x^2+1)^4)/((2x+1)^3tan^5(x^2))]