CRL Exercises 4.17 MrG 2011.1201

3618 days ago by calcpage123

#1) var('a') diff(log(x+a),x) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{a + x}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{a + x}
#3) f(x)=log((1+x^2)/(1-x^2)) show(f(x)) show(diff(f(x),x).simplify_rational()) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\log\left(-\frac{x^{2} + 1}{x^{2} - 1}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{4 \, x}{x^{4} - 1}
\newcommand{\Bold}[1]{\mathbf{#1}}\log\left(-\frac{x^{2} + 1}{x^{2} - 1}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{4 \, x}{x^{4} - 1}
#5) f(x)=log(x^3-2*x+5) show("f(x)=") show(f(x)) show("f'(x)=") show(diff(f(x),x)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f(x)=|
\newcommand{\Bold}[1]{\mathbf{#1}}\log\left(x^{3} - 2 \, x + 5\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f'(x)=|
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{3 \, x^{2} - 2}{x^{3} - 2 \, x + 5}
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f(x)=|
\newcommand{\Bold}[1]{\mathbf{#1}}\log\left(x^{3} - 2 \, x + 5\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f'(x)=|
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{3 \, x^{2} - 2}{x^{3} - 2 \, x + 5}
#7) f(x)=x*log(x) show("f(x)=") show(f(x)) show("f'(x)=") show(diff(f(x),x)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f(x)=|
\newcommand{\Bold}[1]{\mathbf{#1}}x \log\left(x\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f'(x)=|
\newcommand{\Bold}[1]{\mathbf{#1}}\log\left(x\right) + 1
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f(x)=|
\newcommand{\Bold}[1]{\mathbf{#1}}x \log\left(x\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f'(x)=|
\newcommand{\Bold}[1]{\mathbf{#1}}\log\left(x\right) + 1
#9) f(x)=log(x)^3 show("f(x)=") show(f(x)) show("f'(x)=") show(diff(f(x),x)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f(x)=|
\newcommand{\Bold}[1]{\mathbf{#1}}\log\left(x\right)^{3}
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f'(x)=|
\newcommand{\Bold}[1]{\mathbf{#1}}3 \, \frac{1}{x} \log\left(x\right)^{2}
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f(x)=|
\newcommand{\Bold}[1]{\mathbf{#1}}\log\left(x\right)^{3}
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f'(x)=|
\newcommand{\Bold}[1]{\mathbf{#1}}3 \, \frac{1}{x} \log\left(x\right)^{2}
#11) f(x)=log(x+(1+x^2)^(1/2)) show("f'(x)=") show(diff(f(x),x).factor()) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f'(x)=|
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{\sqrt{x^{2} + 1}}
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f'(x)=|
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{\sqrt{x^{2} + 1}}
#13) f(x)=e^(4*x+5) show("f(x)=") show(f(x)) show("f'(x)=") show(diff(f(x),x).factor()) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f(x)=|
\newcommand{\Bold}[1]{\mathbf{#1}}e^{\left(4 \, x + 5\right)}
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f'(x)=|
\newcommand{\Bold}[1]{\mathbf{#1}}4 \, e^{\left(4 \, x + 5\right)}
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f(x)=|
\newcommand{\Bold}[1]{\mathbf{#1}}e^{\left(4 \, x + 5\right)}
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f'(x)=|
\newcommand{\Bold}[1]{\mathbf{#1}}4 \, e^{\left(4 \, x + 5\right)}
#15) f(t)=log(3-2*t^2) show("f(t)=") show(f(t)) show("f'(t)=") show(diff(f(t),t)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f(t)=|
\newcommand{\Bold}[1]{\mathbf{#1}}\log\left(-2 \, t^{2} + 3\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f'(t)=|
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{4 \, t}{2 \, t^{2} - 3}
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f(t)=|
\newcommand{\Bold}[1]{\mathbf{#1}}\log\left(-2 \, t^{2} + 3\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f'(t)=|
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{4 \, t}{2 \, t^{2} - 3}
#17) var('b') f(x)=e^(b^2+x^2) show("f(x)=") show(f(x)) show("f'(x)=") show(diff(f(x),x)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f(x)=|
\newcommand{\Bold}[1]{\mathbf{#1}}e^{\left(b^{2} + x^{2}\right)}
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f'(x)=|
\newcommand{\Bold}[1]{\mathbf{#1}}2 \, x e^{\left(b^{2} + x^{2}\right)}
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f(x)=|
\newcommand{\Bold}[1]{\mathbf{#1}}e^{\left(b^{2} + x^{2}\right)}
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f'(x)=|
\newcommand{\Bold}[1]{\mathbf{#1}}2 \, x e^{\left(b^{2} + x^{2}\right)}
#19) var('b,s') f(s)=b^(s^2) show("f(s)=") show(f(s)) show("f'(s)=") show(diff(f(s),s)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f(s)=|
\newcommand{\Bold}[1]{\mathbf{#1}}b^{\left(s^{2}\right)}
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f'(s)=|
\newcommand{\Bold}[1]{\mathbf{#1}}2 \, b^{\left(s^{2}\right)} s \log\left(b\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f(s)=|
\newcommand{\Bold}[1]{\mathbf{#1}}b^{\left(s^{2}\right)}
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|f'(s)=|
\newcommand{\Bold}[1]{\mathbf{#1}}2 \, b^{\left(s^{2}\right)} s \log\left(b\right)
#39) var('y') psi(y)=log(sqrt((1+y)/(1-y))) show(psi(y)) show(diff(psi(y),y).simplify_rational()) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\log\left(\sqrt{-\frac{y + 1}{y - 1}}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{y^{2} - 1}
\newcommand{\Bold}[1]{\mathbf{#1}}\log\left(\sqrt{-\frac{y + 1}{y - 1}}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{y^{2} - 1}