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What is the value of
(d)/(dx)(x^(-3)) at
x=3 ?

What is the value of $\frac{d}{d x}\left(x^{-3}\right)$ at $x=3$ ?

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Find values of derivatives using limits

Full solution

Q. What is the value of

\frac{d}{d x}\left(x^{-3}\right)

at

x=3

?

Identify Function & Operation: Identify the function and the operation needed. $\newline$ We need to find the derivative of the function $f(x) = x^{-3}$ with respect to $x$ and then evaluate this derivative at $x=3$ .
Apply Power Rule: Apply the power rule for differentiation. $\newline$ The power rule states that the derivative of $x^n$ with respect to $x$ is $n*x^{(n-1)}$ . Applying this rule to our function, we get: $\newline$ $f'(x) = \frac{d}{dx}(x^{-3}) = -3*x^{(-3-1)} = -3*x^{-4}.$
Simplify Derivative Expression: Simplify the expression for the derivative. $f'(x) = -3x^{-4}$ can be rewritten as $f'(x) = -\frac{3}{x^4}$ for easier evaluation.
Evaluate at $x=3$ : Evaluate the derivative at $x=3$ . $\newline$ Substitute $x$ with $3$ in the expression for $f'(x)$ : $\newline$ $f'(3) = -\frac{3}{3^4} = -\frac{3}{81}$ .
Simplify Result: Simplify the result. $\newline$ $-\frac{3}{81}$ simplifies to $-\frac{1}{27}$ .

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