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(d)/(dx)=-(x+1)^(2)

$\frac{d}{d x}=-(x+1)^{2}$

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Simplify radical expressions with root inside the root

Full solution

Q.

\frac{d}{d x}=-(x+1)^{2}

Identify Function: We are asked to find the derivative of the function $f(x) = -(x+1)^{2}$ with respect to $x$ . We will use the power rule and the chain rule for differentiation.
Apply Power Rule and Chain Rule: The power rule states that the derivative of $x^n$ with respect to $x$ is $n*x^{(n-1)}$ . The chain rule states that the derivative of a composite function $f(g(x))$ is $f'(g(x))*g'(x)$ .
Find Derivative of $f(u)$ : Let's apply the chain rule to our function. Let $u = x+1$ , then our function becomes $f(u) = -u^2$ . We will first find the derivative of $f(u)$ with respect to $u$ , which is $f'(u) = -2u$ .
Find Derivative of u: Now we need to find the derivative of $u$ with respect to $x$ , which is $u' = (x+1)' = 1$ , since the derivative of $x$ is $1$ and the derivative of a constant is $0$ .
Apply Chain Rule: Using the chain rule, we multiply the derivative of $f$ with respect to $u$ by the derivative of $u$ with respect to $x$ to get the derivative of $f$ with respect to $x$ . So, the derivative of $f(x)$ with respect to $x$ is $f'(x) = f'(u) \cdot u' = -2u \cdot 1$ .
Substitute $u$ back: Substitute $u$ back into the equation to get the derivative in terms of $x$ . So, $f'(x) = -2(x+1)$ .
Expand the Derivative: We can expand the derivative to get $f'(x) = -2x - 2$ .

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