510 U-Sub MrG 2013.0426 -- Sage

#1) find the derivative of f(x)=x f(x)=x f(x).diff()

 $\newcommand{\Bold}[1]{\mathbf{#1}}1$

#1A) find the antiderivative of f(x)=1 g(x)=1 integrate(g(x),x)

 $\newcommand{\Bold}[1]{\mathbf{#1}}x$

#2) find the derivative of f(x)=x^2 f(x)=x^2 f(x).diff()

 $\newcommand{\Bold}[1]{\mathbf{#1}}2 \, x$

#2A) find the antiderivative of f(x)=2x g(x)=2*x integrate(g(x),x)

 $\newcommand{\Bold}[1]{\mathbf{#1}}x^{2}$

#3) find the derivative of f(x)=x^3 f(x)=x^3 f(x).diff()

 $\newcommand{\Bold}[1]{\mathbf{#1}}3 \, x^{2}$

#3A) find the antiderivative of f(x)=3x^2 g(x)=3*x^2 integrate(g(x),x)

 $\newcommand{\Bold}[1]{\mathbf{#1}}x^{3}$

#4) find the derivative of f(x)=sin(x^2) f(x)=sin(x^2) f(x).diff()

 $\newcommand{\Bold}[1]{\mathbf{#1}}2 \, x \cos\left(x^{2}\right)$

#4A) find the antiderivative of f(x)=xcos(x^2) g(x)=x*cos(x^2) integrate(g(x),x)

 $\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2} \, \sin\left(x^{2}\right)$

#5) find the derivative of f(x)=cos(x^3) f(x)=cos(x^3) f(x).diff()

 $\newcommand{\Bold}[1]{\mathbf{#1}}-3 \, x^{2} \sin\left(x^{3}\right)$

#5A) find the antiderivative of f(x)=x^2*sin(x^3) g(x)=x^2*sin(x^3) integrate(g(x),x)

 $\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{3} \, \cos\left(x^{3}\right)$

#6) find the derivative of f(x)=tan(3*x) f(x)=tan(3*x) f(x).diff()

 $\newcommand{\Bold}[1]{\mathbf{#1}}3 \, \tan\left(3 \, x\right)^{2} + 3$

#6A) find the antiderivative of f(x)=sec^2(3*x) g(x)=(1/cos(3*x)^2) integrate(g(x),x)

 $\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{3} \, \tan\left(3 \, x\right)$

#7) find the derivative of f(x)=e^(x/2) f(x)=exp(x/2) f(x).diff()

 $\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2} \, e^{\left(\frac{1}{2} \, x\right)}$

#7A) find the antiderivative of f(x)=e^(x/2) g(x)=(exp(x/2)) integrate(g(x),x)

 $\newcommand{\Bold}[1]{\mathbf{#1}}2 \, e^{\left(\frac{1}{2} \, x\right)}$

#8) find the derivative of f(x)=ln(sin(x)) f(x)=log(sin(x)) f(x).diff()

 $\newcommand{\Bold}[1]{\mathbf{#1}}\frac{\cos\left(x\right)}{\sin\left(x\right)}$

#8A) find the antiderivative of f(x)=cot(x) g(x)=cot(x) integrate(g(x),x)

 $\newcommand{\Bold}[1]{\mathbf{#1}}\log\left(\sin\left(x\right)\right)$

#9) find the derivative of f(x)=ln(sin(sqrt(x))) f(x)=log(sin(sqrt(x))) f(x).diff()

 $\newcommand{\Bold}[1]{\mathbf{#1}}\frac{\cos\left(\sqrt{x}\right)}{2 \, \sqrt{x} \sin\left(\sqrt{x}\right)}$

#9A) find the antiderivative of f(x)=cot(sqrt(x))/sqrt(x) g(x)=cot(sqrt(x))/sqrt(x) integrate(g(x),x)

 $\newcommand{\Bold}[1]{\mathbf{#1}}2 \, \log\left(\sin\left(\sqrt{x}\right)\right)$

#10) find the derivative of f(x)=exp((sqrt(x))) f(x)=exp((sqrt(x))) f(x).diff()

 $\newcommand{\Bold}[1]{\mathbf{#1}}\frac{e^{\sqrt{x}}}{2 \, \sqrt{x}}$

#10A) find the antiderivative of f(x)=exp(sqrt(x))/sqrt(x) g(x)=exp(sqrt(x))/sqrt(x) integrate(g(x),x)

 $\newcommand{\Bold}[1]{\mathbf{#1}}2 \, e^{\sqrt{x}}$

510 U-Sub MrG 2013.0426