# Taylor series with 3 variables

Hi am trying to solve this Taylor series with 3 variables but my result is not equal to the solution- So i think i might be wrong expanding the taylor series, or the solution is not correct

1. Homework Statement
Find an a approximated value for the function f(x,y,z) = 2x + ( 1 + y) * sin z at the point
x= 0.1 y = 0.2 z = 0.3 [asnwer = 0.5546]

Use only linear terms ( dx dy dz has only 1 power)

2. Homework Equations

df = dx*$$\frac{f'(f)}{f'(x)}$$ + dy*$$\frac{f'(f)}{f'(y)}$$ + dz *$$\frac{f'(f)}{f'(z)}$$ + ...

3. The Attempt at a Solution
So i started evaluating the function at the point (0,0,0) which gives me the value 0. Then i expand the taylor series like this

df = dx *$$\frac{f'(f)}{f'(x)}$$ + dy *$$\frac{f'(f)}{f'(y)}$$ + dz *$$\frac{f'(f)}{f'(z)}$$ + dxdy *$$\frac{f''(f)}{f'(x)f'(y) }$$ + dxdz *$$\frac{f'(f)}{f''(x)f'(z)}$$ + dydz *$$\frac{f''(f)}{f'(y)f'(z)}$$ + dxdydz $$\frac{f'''(f)}{f'(x)f'(y)f'(z)}$$

Then i calculate the differential equations
$$\frac{f'(f)}{f'(x)}$$ = 2
$$\frac{f'(f)}{f'(y)}$$ = sin z
$$\frac{f'(f)}{f'(z)}$$ = (1 + y) cos z
$$\frac{f''(f)}{f'(x)f'(y) }$$ = 0
$$\frac{f'(f)}{f''(x)f'(z)}$$ = 0
$$\frac{f''(f)}{f'(y)f'(z)}$$ = cos z
$$\frac{f'''(f)}{f'(x)f'(y)f'(z)}$$ = 0

So my final formula is

df = 2 dx + dz * cos(0) + dydz * cos (0)

An evaluating that formula in (0.1,0.2,0.3)

df(0.1,0.2,0.3) = 2 *0.1 + 0.3 + 0.06 = 0.56

But the solution given by the exercise is 0.5546

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