Differentiation

• Feb 19th 2009, 04:03 PM
calcbeg
Differentiation
Hi

I would appreciate if someone could check my work.

Differentiate the following functions

a)
f(x) = ln (tan 3x + cos 5x)^2

f'(x) = (1/(tan 3x + cos 5x)^2) (2(sec^2^x + -sin 5x))

f'(x) = (2sec^2 3x - 2sin 5x)/((tan 3x + cos 5x)^2)

b)
f(x) = e^x^3 + 2ln x) cos (1/x^2))

f'(x) = (e^x^3 + 2 ln x) ((-sin (1/x^2))(-2x)) + (e^x^3 + 2/x) (cos (1/x^2))

c)
f(x) = sin ^ tan x

ln f(x) = ln (sin ^ tanx)
ln f(x) = tanx (ln sin)
f'(x) (1/f(x)) = ((sec^2 x) (ln sin) + (1/sin)(tanx))
f'(x) = sin ^tan x ((sec^2x)(ln sin) + (1/sin)(tan x))

• Feb 19th 2009, 04:51 PM
HallsofIvy
Quote:

Originally Posted by calcbeg
Hi

I would appreciate if someone could check my work.

Differentiate the following functions

a)
f(x) = ln (tan 3x + cos 5x)^2

f'(x) = (1/(tan 3x + cos 5x)^2) (2(sec^2^x + -sin 5x))

That should be 2(3sec^3(3x)- 5sin(5x))

Quote:

f'(x) = (2sec^2 3x - 2sin 5x)/((tan 3x + cos 5x)^2)

b)
f(x) = e^x^3 + 2ln x) cos (1/x^2))

f'(x) = (e^x^3 + 2 ln x) ((-sin (1/x^2))(-2x)) + (e^x^3 + 2/x) (cos (1/x^2))
What happened to the derivative of e^(x^3)? rather that the last e^x^3, you shouled have (e^(x^3))(2x^2).

Quote:

c)
f(x) = sin ^ tan x

ln f(x) = ln (sin ^ tanx)
ln f(x) = tanx (ln sin)
f'(x) (1/f(x)) = ((sec^2 x) (ln sin) + (1/sin)(tanx))
f'(x) = sin ^tan x ((sec^2x)(ln sin) + (1/sin)(tan x))