CRL 3.8 Odds MrG Defn of Derivative

3682 days ago by calcpage123

#1) var('h') f(x)=3*x^2 show((f(x+h)-f(x))/h) show(((f(x+h)-f(x))).expand()/h) show(((f(x+h)-f(x))/h).expand()) show(limit((f(x+h)-f(x))/h,h=0)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{3 \, {\left({\left(h + x\right)}^{2} - x^{2}\right)}}{h}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{3 \, {\left(h^{2} + 2 \, h x\right)}}{h}
\newcommand{\Bold}[1]{\mathbf{#1}}3 \, h + 6 \, x
\newcommand{\Bold}[1]{\mathbf{#1}}6 \, x
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{3 \, {\left({\left(h + x\right)}^{2} - x^{2}\right)}}{h}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{3 \, {\left(h^{2} + 2 \, h x\right)}}{h}
\newcommand{\Bold}[1]{\mathbf{#1}}3 \, h + 6 \, x
\newcommand{\Bold}[1]{\mathbf{#1}}6 \, x
show(diff(f(x),x)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}6 \, x
\newcommand{\Bold}[1]{\mathbf{#1}}6 \, x
#3) f(x)=5-4*x show(limit((f(x+h)-f(x))/h,h=0)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}-4
\newcommand{\Bold}[1]{\mathbf{#1}}-4
#5) f(x)=1/x show(limit((f(x+h)-f(x))/h,h=0)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{x^{2}}
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{x^{2}}
#7) f(x)=x^3 show(limit((f(x+h)-f(x))/h,h=0)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}3 \, x^{2}
\newcommand{\Bold}[1]{\mathbf{#1}}3 \, x^{2}
#9) f(x)=1-2*x^3 show(limit((f(x+h)-f(x))/h,h=0)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}-6 \, x^{2}
\newcommand{\Bold}[1]{\mathbf{#1}}-6 \, x^{2}
#11) f(x)=2/x^2 show(limit((f(x+h)-f(x))/h,h=0)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{4}{x^{3}}
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{4}{x^{3}}
#13) f(x)=7*x^2+x show(limit((f(x+h)-f(x))/h,h=0)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}14 \, x + 1
\newcommand{\Bold}[1]{\mathbf{#1}}14 \, x + 1
#15) var('t') r(t)=8*t+3*t^2 show(limit((r(t+h)-r(t))/h,h=0)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}6 \, t + 8
\newcommand{\Bold}[1]{\mathbf{#1}}6 \, t + 8
#17) var('a') s(t)=-a/(2*t+3) show(limit((s(t+h)-s(t))/h,h=0)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{2 \, a}{4 \, t^{2} + 12 \, t + 9}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{2 \, a}{4 \, t^{2} + 12 \, t + 9}
#19) f(x)=3*x^3-2*x^2 show(limit((f(x+h)-f(x))/h,h=0)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}9 \, x^{2} - 4 \, x
\newcommand{\Bold}[1]{\mathbf{#1}}9 \, x^{2} - 4 \, x
#21) f(x)=(x^2-5)/x show(limit((f(x+h)-f(x))/h,h=0)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{x^{2} + 5}{x^{2}}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{x^{2} + 5}{x^{2}}
#XTRA) f(x)=cos(x) show((f(x+h)-f(x))/h) show(((f(x+h)-f(x))/h).expand_trig()) show(limit((f(x+h)-f(x))/h,h=0)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{\cos\left(h + x\right) - \cos\left(x\right)}{h}
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{\sin\left(h\right) \sin\left(x\right) - \cos\left(h\right) \cos\left(x\right) + \cos\left(x\right)}{h}
\newcommand{\Bold}[1]{\mathbf{#1}}-\sin\left(x\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{\cos\left(h + x\right) - \cos\left(x\right)}{h}
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{\sin\left(h\right) \sin\left(x\right) - \cos\left(h\right) \cos\left(x\right) + \cos\left(x\right)}{h}
\newcommand{\Bold}[1]{\mathbf{#1}}-\sin\left(x\right)