How Do You Find the Derivative of the Function f(x) = pi*x + 1/(cos^2(Pi*x))?

[PrudensOptimus]

Find the derivative for the f(x).
f(x) = pi*x + 1/(cos^2(Pi*x))
And find f'(1/4)

Here's what i got.
f'(x) = pi * d/dx (x) + d/dx (cos^2 pi*x)^-1
= pi + [ (- (d/dx cos^2 (pi*x))/(cos^4(pi*x)) ]
= pi + [ (- (d/dx cos(pi*x) * (2 * cos(pi * x))/(cos^4(pi*x)) ]
= pi + [ (pi * sin x * 2cos(pi * x)) / ... ]
= pi + [ (2 * pi * sin x) / cos^3 (pi * x) ]

Thus f'(1/4) = 7.5251

 

[PrudensOptimus]

#2 question:

Find f'(x) and f'(-1).

f(x) = (((1/x^2) - 2)/(1/x^2))²

f'(x) = d/dx ((1/x^2) - 2)/(1/x^2) * (2 * ((1/x^2) - 2)/(1/x^2))
= [2((1/x^2) - 2)/(1/x^2)] * [((1/x^2)(-2/x^3) - ((1/x^2)-2)(-2/x^3))/((1/x^2)^2)]
= ... * [ -4/x^7 ]
= -8(1 - 2x^2)/x^7

Thus, f'(-1) = -8.

Please correct my mistakes. Thanks.

 

[quantumdude]

Originally posted by PrudensOptimus
Find the derivative for the f(x).
f(x) = pi*x + 1/(cos^2(Pi*x))
And find f'(1/4)
.
.
.
= pi + [ (2 * pi * sin x) / cos^3 (pi * x) ]

Thus f'(1/4) = 7.5251

I don't have a calculator handy, but your derivative is correct.

Find f'(x) and f'(-1).

f(x) = (((1/x^2) - 2)/(1/x^2))²
.
.
.
= -8(1 - 2x^2)/x^7

Thus, f'(-1) = -8.

For the derivative, I get

f'(x)=-8x+16x³

I rewrote the function as follows:

f(x)=(1/x²-2)²x⁴

To get the last factor, I noted that 1/(1/x²) is simply x². Squaring that gives x⁴.

edit: It would be best to expand the function before differentiating, as follows:

f(x)=(1/x⁴-4/x²+4)x⁴

Now distribute the x⁴

f(x)=(1-4x²+4x⁴)

Doesn't look as nasty now, does it?

 

[PrudensOptimus]

You sure you got your #2 derivative right?

It took me two times to re-derive it and everytime I get the same answer.

 

[suffian]

Originally posted by PrudensOptimus
Here's what i got.
f'(x) = pi * d/dx (x) + d/dx (cos^2 pi*x)^-1
= pi + [ (- (d/dx cos^2 (pi*x))/(cos^4(pi*x)) ]
= pi + [ (- (d/dx cos(pi*x) * (2 * cos(pi * x))/(cos^4(pi*x)) ]
= pi + [ (pi * sin x * 2cos(pi * x)) / ... ]
= pi + [ (2 * pi * sin x) / cos^3 (pi * x) ]

Thus f'(1/4) = 7.5251

Should be this on the fourth line:
= π + π sin(π x) 2cos(π x)/cos⁴(π x)
i.e. Missed a π inside the sin(x).

And the final answer:
f'(x) = π + 2π sec²(π x)tan(πx)
f'(1/4) ≈ 15.708

For your second question, you surprisingly did get the correct value for f'(-1), but the function you found is in general wrong:
f(x) = [(1/x²-2) / (1/x²)]²
f(x) = [(1/x²-2) *x²]²
f(x) = (1/x²-2)²*x²²
f(x) = (1/x⁴-4/x²+4)x⁴
f(x) = 1-4x²+4x⁴
(as Tom derived)

From this it far easier to find f'(x):
f'(x)= -8x+16x³
f'(x) = 8x(2x²-1)
f'(-1) = -8
This is similar to be not the same as the function you derived.