Question

3. Find the indicated derivatives of the following functions without inte- grating. Make sure to fully justify your work! () ) (a) Let c E R da cos dt er+1 d tan(t) df (b) dr arctan(r) rt (c) H(r) 1+2+ dt; H"()

Answer 1

PART(a):

f (x) cos2 (t/3) dt COs

ce d cos2 (t/) dt f(x) dr

diff wrt x using FTC

f'(z) cos (e/)e)-cos2 (z/)(2) dz COS da

'(x) cos2 (e) (0)-cos (1) (1) 1/3 COS

1/3 f'(x) cos2 (x/3

PART(b):

tant dt f(x) Jtan-1

$f'(x) =\frac{\mathrm{d} }{\mathrm{d} x}\left [ \int_{\tan^{-1}x}^{x+1} e^{\tan t} dt \right ]$

diff wrt x using FTC

f'(x) =etan (r+1)(1)-etan tan dz (tan da

f(xean(r+ (1) e 2

tan (r+1) e f'(x)e 1 r2

PART(c):

dt 1 t2t Н () т2

xt H'(r) dt 1 t2t dar

diff wrt x using FTC

$H'(x) = \frac{x(-1)}{1+(-1)^{2}+(-1)^{4}}\frac{\mathrm{d} }{\mathrm{d} x}(-1)- \frac{x(x^{2})}{1+(x^{2})^{2}+(x^{2})^{4}}\frac{\mathrm{d} }{\mathrm{d} x}(x^{2})$

H' (x) (0) (2r) 111 1

24 H'(r)0 +4+

$H'(x) =-\frac{2x^{4}}{1+x^{4}+x^{8}}$

diff wrt x using the quotient rule

$H''(x) =-\left [ \frac{(1+x^{4}+x^{8}) (8x^{3})-(4x^{3}+8x^{7}) (2x^{4})}{(1+x^{4}+x^{8})^{2}} \right ]$

3 -8 8 H"(x) 4 (1

$H''(x) = \frac{8x^{11}-8x^3}{(1+x^{4}+x^{8})^{2}}$

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