Question

Oblicz pochodne funkcji w podanym punkcie x.
a) f(x)=5x^7/coś(7x) w punkcie x=π
b)f(x)=ln(6x^2+46x) w punkcie x=4
c) r(x)=sin(4x)e^6x w punkcie x=x/4

Answer 1

[tex]f(x)=\frac{5x^{7} }{cos(7x)} \\f'(x)=\frac{35x^{6}*cos(7x)+(-7sin7x)*5x^{7} }{cos^{2} (7x)} =\frac{35x^{6} *cos(7x)-35x^{7} sin(7x)}{cos^{7x} } =\frac{35x^{6} (cos7x-xsin7x)}{cos^{2}(7x) }[/tex][tex]f'(\pi )=\frac{35\pi ^{6}(cos7\pi -\pi sin7\pi ) }{cos^{2}(7\pi ) } =\frac{35\pi ^{6} (-1-\pi *0)}{(-1)^{2} } =-35\pi ^{6\\}[/tex]

[tex]g(x)=ln(6x^{2} +46x)\\g'(x)=\frac{1}{(6x^{2} +46x)} *(12x+46)\\g'(4)=\frac{1}{6*4^{2}+46*4 } *(12*4+46)=\frac{1}{280} *94=\frac{47}{140}[/tex]

[tex]r(x)=sin(4x)*e^{6x} \\r'(x)=4cos(4x)*e^{6x} +6e^{6x} *sin(4x)\\r'(\frac{\pi }{4} )=4cos\pi *e^{(6\pi)/4 } +6e^{(6*\frac{\pi }{4}) } *sin\pi \\r'(\frac{\pi }{4} )=4*(-1)*e^{\frac{3}{2} \pi} +0\\r'(\frac{\pi }{4} )=-4e^{\frac{3}{2}\pi }[/tex]