Question

if y= 5tanx, find dy/dx at x=3.14​

Answer 1

Answer:

y=$5^{tan x}$,

Multiply both sides by log, [since 5 has the power of tan x over it, after multiplying by log the power comes down]

㏑y=㏑($5^{tanx}$)

㏑y=tanx.㏑5

Now, Differentiate it w.r.t x,

$\frac{d(lny)}{dx\\}$=$\frac{d(tanxln(5))}{dx}$

$\frac{1}{y}$$\frac{dy}{dx}$=㏑5.$sec^{2}x$

$\frac{dy}{dx}$=y($sec^{2}x$.㏑5)

$\frac{dy}{dx}$=$5^{tanx}$($sec^{2}x$.㏑5)

At x=$\frac{\pi }{4}$,

$\frac{dy}{dx}$=$5^{tan\frac{\pi }{4} }$($sec^{2}\frac{\pi }{4}$.㏑5)

$\frac{dy}{dx}$=5(2×1.61)  [$tan\frac{\pi }{4}$=1,$sec^{2}\frac{\pi }{4}$=2,㏑5≅1.61]

$\frac{dy}{dx}$=16.1[approximately 16]

⇒Trick:

$\frac{d[a^{b}] }{dx}$=$a^{b}$.㏑a.$\frac{d[b]}{dx}$