```python
#source https://www.emathhelp.net/notes/calculus-1/differentials/using-differentials-to-estimate-errors/
from IPython.display import Image
from IPython.core.display import HTML 
from sympy import *
Image(url= "https://i.imgur.com/08xREPc.png")
```









```python
L  = Function("L")
f = Function("f")
x,a,Delta,y = symbols('x a Delta y')
Eq(L(x) , f(a) + Derivative(f(a),a)*(x-a))
```




$\displaystyle L{\left(x \right)} = \left(- a + x\right) \frac{d}{d a} f{\left(a \right)} + f{\left(a \right)}$




```python
expr = pi*x**2
x1 = 25
dx1 = 0.3
Eq(f(x),expr)
```




$\displaystyle f{\left(x \right)} = \pi x^{2}$




```python
Eq(Delta*y, f(Delta*x+x) - f(x))
```




$\displaystyle \Delta y = - f{\left(x \right)} + f{\left(\Delta x + x \right)}$




```python
eq0 = Eq(Delta*y, expr.subs(x,x1+dx1) - expr.subs(x,x1))
eq0
```




$\displaystyle \Delta y = 15.09 \pi$




```python
dy = round(eq0.rhs.evalf(),10)
dy
```




$\displaystyle 47.4066331427$




```python
dy,dx = symbols('dy dx')
Eq(dy/dx,Derivative(expr))
```




$\displaystyle \frac{dy}{dx} = \frac{d}{d x} \pi x^{2}$




```python
expr2 = solve(Eq((dy/dx),diff(expr)),dy)[0]
Eq(dy,expr2)
```




$\displaystyle dy = 2 \pi dx x$




```python
dy = expr2.subs(dx,dx1).subs(x,x1).evalf()
dy

```




$\displaystyle 47.1238898038469$




```python
dy/expr.subs(x,x1).evalf()
```




$\displaystyle 0.024$




```python
0.024*100
```




    2.4




```python
Image(url= "https://i.imgur.com/Pp7KEKe.png")
```








c3739ef24ff39b125fe1d207c63ad6235cc2dde9
ipynb
Jupyter Notebook
Calculus_Homework/WWB15.11.ipynb
NSC9/Sample_of_Work
8f8160fbf0aa4fd514d4a5046668a194997aade6
MIT
Calculus_Homework/WWB15.11.ipynb
NSC9/Sample_of_Work
8f8160fbf0aa4fd514d4a5046668a194997aade6
MIT
Calculus_Homework/WWB15.11.ipynb
NSC9/Sample_of_Work
8f8160fbf0aa4fd514d4a5046668a194997aade6
MIT
Qwen/Qwen-72B
1. YES
2. YES