hello i need help for derivate this function: x+1+1/(x+2)
thanks for your help
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let y = x +1 + 1/(x+2)
= x + 1 + (x+2)^-1
dy/dx= 1 - (x+2)^-2
or
= 1 - 1/(x+2)^2
To find the derivative of the given function, we can use the rules of differentiation. Here's how you can do it step by step:
Step 1: Identify the function you want to differentiate. In this case, the function is f(x) = x + 1 + 1/(x + 2).
Step 2: Start by differentiating each term separately.
- Differentiating the term "x" gives us 1.
- Differentiating the term "1" gives us 0 because it's a constant.
- Differentiating the term "1/(x + 2)" requires the quotient rule.
Step 3: Apply the quotient rule to differentiate the term "1/(x + 2)".
The quotient rule states that if we have a function of the form f(x)/g(x), then its derivative can be found using the formula:
[f'(x)g(x) - f(x)g'(x)] / [g(x)]^2.
In this case, f(x) = 1 and g(x) = (x + 2).
- Differentiating f(x) = 1 gives us f'(x) = 0 (since it's a constant).
- Differentiating g(x) = (x + 2) gives us g'(x) = 1.
Plugging these values into the quotient rule formula, we get:
[0(x + 2) - 1(1)] / (x + 2)^2
= -1 / (x + 2)^2.
Step 4: Sum up the derivatives of each term.
Adding up the derivatives of each term, we get:
1 + 0 - 1 / (x + 2)^2
= 1 - 1 / (x + 2)^2.
Therefore, the derivative of the function f(x) = x + 1 + 1/(x + 2) is f'(x) = 1 - 1 / (x + 2)^2.