Question

Find derivative using first principle definition f(x)=(square root of 9-x)+ ( -2x+1/(x+2))

Answer 1

Step-by-step explanation:

Here we have the following function:

`f(x)=√(9-x)+(-2x+1)/(x+2)`

By property:

`\left(f\pm g\right)'=f\:'\pm g'`

Then:

`(df)/(dx)=(d)/(dx)\left(√(9-x)\right)+(d)/(dx)\left((1-2x)/(x+2)\right)`

From here:

`(d)/(dx)\left(√(9-x)\right)=-(1)/(2√(9-x))`

And:

`(d)/(dx)\left((1-2x)/(x+2)\right)=((d)/(dx)[\left(1-2x\right)]\left(x+2\right)-(d)/(dx)[\left(x+2\right)]\left(1-2x\right))/(\left(x+2\right)^2) \\ \\ =(\left(-2\right)\left(x+2\right)-1\cdot \left(1-2x\right))/(\left(x+2\right)^2)= -(5)/(\left(x+2\right)^2)`

Therefore:

`\boxed{(df)/(dx)=-(1)/(2√(9-x))-(5)/(\left(x+2\right)^2)}`