(d)/(dx)(1)/((x+5)^(2))

Identify Function: We need to find the derivative of the function f(x) = 1/((x+5)^(2)) with respect to x. This is a problem involving the derivative of a function in the form of a quotient where the numerator is a constant and the denominator is a variable raised to a power. We will use the chain rule and the power rule to find the derivative.Rewrite Function: Let's rewrite the function to make it easier to differentiate. We can write the function as f(x) = (x+5)^(-2). This is equivalent to the original function but is now in a form that is easier to differentiate using the power rule.Apply Power Rule: Now we apply the power rule for differentiation, which states that the derivative of x^n with respect to x is n*x^(n-1). In this case, we have a function of x (x+5) raised to the power of -2. So, we will differentiate (x+5)^(-2) as if it were x^n.Calculate Derivative: Taking the derivative, we get f'(x) = -2*(x+5)^(-2-1) = -2*(x+5)^(-3). We have multiplied the exponent by the coefficient and subtracted one from the exponent, following the power rule.Rewrite Final Result: Finally, we can rewrite the derivative in a more readable form. The derivative f'(x) = -2*(x+5)^(-3) can be written as f'(x) = -2/((x+5)^(3)).

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$\frac{d}{d x} \frac{1}{(x+5)^{2}}$

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Algebra 1

Multiplication with rational exponents

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\frac{d}{d x} \frac{1}{(x+5)^{2}}

Identify Function: We need to find the derivative of the function $f(x) = \frac{1}{(x+5)^{2}}$ with respect to $x$ . This is a problem involving the derivative of a function in the form of a quotient where the numerator is a constant and the denominator is a variable raised to a power. We will use the chain rule and the power rule to find the derivative.
Rewrite Function: Let's rewrite the function to make it easier to differentiate. We can write the function as $f(x) = (x+5)^{-2}$ . This is equivalent to the original function but is now in a form that is easier to differentiate using the power rule.
Apply Power Rule: Now we apply the power rule for differentiation, which states that the derivative of $x^n$ with respect to $x$ is $n*x^{(n-1)}$ . In this case, we have a function of $x$ ( $x+5$ ) raised to the power of $-2$ . So, we will differentiate $(x+5)^{-2}$ as if it were $x^n$ .
Calculate Derivative: Taking the derivative, we get $f'(x) = -2\cdot(x+5)^{-2-1} = -2\cdot(x+5)^{-3}$ . We have multiplied the exponent by the coefficient and subtracted one from the exponent, following the power rule.
Rewrite Final Result: Finally, we can rewrite the derivative in a more readable form. The derivative $f'(x) = -2\cdot(x+5)^{-3}$ can be written as $f'(x) = -\frac{2}{((x+5)^{3})}$ .