Value of the differential of tan(x)^cos(x) at pi/4

[ishant]

Homework Statement

^

Homework Equations

The Attempt at a Solution

y = tanx^cosx

lny = cosx(ln(tanx))

1/y(dy/dx) = (sec^2x * cosx)/tanx + ln(tanx)(-sinx)

at pi/4

1/y(dy/dx) = (1 * 1/sqrt(2))/1 + ln(1)(-sin(pi/4))

1/y(dy/dx) = 1/sqrt(2) + 0 ******* ln1 = 0

***
y = tanx^cosx => 1^(1/sqrt(2)) = 1 => 1/y = 1
***
dy/dx at pi/4 = 1/sqrt(2)

The answer is sqrt(2) NOT 1/sqrt(2)

What am I doing wrong?

 

[SammyS]

Homework Statement

Homework Equations

The Attempt at a Solution

y = tanx^cosx

lny = cosx(ln(tanx))

1/y(dy/dx) = (sec^2x * cosx)/tanx + ln(tanx)(-sinx)

at pi/4

1/y(dy/dx) = (1 * 1/sqrt(2))/1 + ln(1)(-sin(pi/4))

1/y(dy/dx) = 1/sqrt(2) + 0 ******* ln1 = 0

***
y = tanx^cosx => 1^(1/sqrt(2)) = 1 => 1/y = 1
***
dy/dx at pi/4 = 1/sqrt(2)

The answer is sqrt(2) NOT 1/sqrt(2)

What am I doing wrong?

Hello ishant. Welcome to PF !

What is sec²(π/4) ?

 

[ishant]

Hello ishant. Welcome to PF !

What is sec²(π/4) ?

Oh my god! Thank you!