Question

d/dx(1/x^4secx) differentiate

Answer 1

GIVEN :

The differential expression is $\frac{d}{dx}(\frac{1}{x^4secx})$

TO FIND :

The derivative of the given the differential expression

SOLUTION:

Given that the differential expression is $\frac{d}{dx}(\frac{1}{x^4secx})$

By using the Formulae in differentiation :

i) $\frac{d}{dx}(\frac{u}{v})=\frac{v.\frac{du}{dx}-u.\frac{dv}{dx}}{v^2}$

ii) $\frac{d}{dx}(uv)=u.\frac{dv}{dx}+v.\frac{du}{dx}$ and

iii) $\frac{d}{dx}(x^n)=n.x^{n-1}$

$\frac{d}{dx}(\frac{1}{x^4secx})$

Now differentiating the above equation with respect to x we get

$=\frac{x^4secx.\frac{d(1)}{dx}-1.\frac{d}{dx}(x^4secx)}{(x^4secx)^2}$

By using the differentiation formula,

$\frac{d}{dx}(c)=0$ where c is a constant.

$=\frac{x^4secx.(0)-1.\frac{d}{dx}(x^4secx)}{(x^4secx)^2}$

$=\frac{0-(x^4.(secxtanx)+secx.(4x^{4-1}))}{x^8sec^2x}$

By using the differentiation formula,

$\frac{d}{dx}(secx)=secxtanx$

$=-\frac{x^4.(secxtanx)+secx.(4x^3)}{x^8sec^2x}$

$=\frac{-4x^3secx-x^4secxtanx}{x^8sec^2x}$

$=\frac{-4x^3secx}{x^8sec^2x}-\frac{x^4secxtanx}{x^8sec^2x}$

$=-\frac{4}{x^8.x^{-3}secx}-\frac{tanx}{x^8.x^{-4}secx}$

$=-\frac{4}{x^{8-3}secx}-\frac{tanx}{x^{8-4}secx}$

$=-\frac{4}{x^{5}secx}-\frac{tanx}{x^{4}secx}$

∴ $\frac{d}{dx}(\frac{1}{x^4secx})=-\frac{4}{x^{5}secx}-\frac{tanx}{x^{4}secx}$

∴ the differentiation of the given expression $\frac{d}{dx}(\frac{1}{x^4secx})$  is $-\frac{4}{x^{5}secx}-\frac{tanx}{x^{4}secx}$

Answer 2

Answer:

Step-by-step explanation: