CRL 4.1 MrG Differentiation Rules

4228 days ago by calcpage123

# Product Rule f(x)=x^2 g(x)=sin(x) diff(f(x)*g(x),x) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}x^{2} \cos\left(x\right) + 2 \, x \sin\left(x\right)
\newcommand{\Bold}[1]{\mathbf{#1}}x^{2} \cos\left(x\right) + 2 \, x \sin\left(x\right)
f(x)=function('f',x) g(x)=function('g',x) diff(f(x)*g(x),x) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}f\left(x\right) D[0]\left(g\right)\left(x\right) + g\left(x\right) D[0]\left(f\right)\left(x\right)
\newcommand{\Bold}[1]{\mathbf{#1}}f\left(x\right) D[0]\left(g\right)\left(x\right) + g\left(x\right) D[0]\left(f\right)\left(x\right)
# Quotient Rule diff(f(x)/g(x),x) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{f\left(x\right) D[0]\left(g\right)\left(x\right)}{g\left(x\right)^{2}} + \frac{D[0]\left(f\right)\left(x\right)}{g\left(x\right)}
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{f\left(x\right) D[0]\left(g\right)\left(x\right)}{g\left(x\right)^{2}} + \frac{D[0]\left(f\right)\left(x\right)}{g\left(x\right)}
diff(f(x)/g(x),x).simplify_rational() 
        
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{f\left(x\right) D[0]\left(g\right)\left(x\right) - g\left(x\right) D[0]\left(f\right)\left(x\right)}{g\left(x\right)^{2}}
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{f\left(x\right) D[0]\left(g\right)\left(x\right) - g\left(x\right) D[0]\left(f\right)\left(x\right)}{g\left(x\right)^{2}}
(sin(x)^2+cos(x)^2).simplify_trig() 
        
\newcommand{\Bold}[1]{\mathbf{#1}}1
\newcommand{\Bold}[1]{\mathbf{#1}}1
var('y') sin(x+y).simplify_trig() 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(x\right) \cos\left(y\right) + \sin\left(y\right) \cos\left(x\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(x\right) \cos\left(y\right) + \sin\left(y\right) \cos\left(x\right)
simplify? 
       

File: /home/medlock/sage-4.7.1/local/lib/python2.6/site-packages/sage/calculus/functional.py

Type: <type ‘function’>

Definition: simplify(f)

Docstring:

Simplify the expression f.

EXAMPLES: We simplify the expression i + x - x.

sage: f = I + x - x; simplify(f)
I

In fact, printing f yields the same thing - i.e., the simplified form.

File: /home/medlock/sage-4.7.1/local/lib/python2.6/site-packages/sage/calculus/functional.py

Type: <type ‘function’>

Definition: simplify(f)

Docstring:

Simplify the expression f.

EXAMPLES: We simplify the expression i + x - x.

sage: f = I + x - x; simplify(f)
I

In fact, printing f yields the same thing - i.e., the simplified form.
# Power Rule and Chain Rule var('n') diff((f(x))^n,x) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}n f\left(x\right)^{{\left(n - 1\right)}} D[0]\left(f\right)\left(x\right)
\newcommand{\Bold}[1]{\mathbf{#1}}n f\left(x\right)^{{\left(n - 1\right)}} D[0]\left(f\right)\left(x\right)
diff((x^3)^n,x) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}3 \, n x^{2} {\left(x^{3}\right)}^{{\left(n - 1\right)}}
\newcommand{\Bold}[1]{\mathbf{#1}}3 \, n x^{2} {\left(x^{3}\right)}^{{\left(n - 1\right)}}
(diff(sqrt(x^3),x)) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{3 \, x^{2}}{2 \, \sqrt{x^{3}}}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{3 \, x^{2}}{2 \, \sqrt{x^{3}}}
(diff(sqrt(x^3),x)).simplify_radical() 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{3}{2} \, \sqrt{x}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{3}{2} \, \sqrt{x}
diff((sin(x))^3,x) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}3 \, \sin\left(x\right)^{2} \cos\left(x\right)
\newcommand{\Bold}[1]{\mathbf{#1}}3 \, \sin\left(x\right)^{2} \cos\left(x\right)
g(x)=diff((sin(x))^3,x) g(0) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}0
\newcommand{\Bold}[1]{\mathbf{#1}}0