Let g(x)=(1)/(x^(3)). g^(')(3)=

Identify Function: Identify the function to differentiate. We are given the function g(x) = 1/x^3, and we need to find its derivative, g'(x). Differentiate Using Power Rule: Differentiate the function using the power rule. The power rule states that the derivative of x^n is n*x^(n-1). In this case, we can rewrite the function as g(x) = x^(-3) and then differentiate it. g'(x) = -3*x^(-3-1) = -3*x^(-4) Evaluate at x = 3: Evaluate the derivative at x = 3. Now we substitute x = 3 into the derivative to find g'(3). g'(3) = -3*(3)^(-4) Simplify Expression: Simplify the expression. g'(3) = -3*(1/3^4) = -3*(1/81) = -3/81 Reduce Fraction: Reduce the fraction to its simplest form. -3/81 can be simplified by dividing both the numerator and the denominator by 3. g'(3) = -1/27

Home

Math Problems

Algebra 2

Simplify variable expressions using properties

Full solution

Q. Let

g(x)=\frac{1}{x^{3}}

\newline

g^{\prime}(3)=

Identify Function: Identify the function to differentiate. $\newline$ We are given the function $g(x) = \frac{1}{x^3}$ , and we need to find its derivative, $g'(x)$ .
Differentiate Using Power Rule: Differentiate the function using the power rule. $\newline$ The power rule states that the derivative of $x^n$ is $n*x^{(n-1)}$ . In this case, we can rewrite the function as $g(x) = x^{-3}$ and then differentiate it. $\newline$ $g'(x) = -3*x^{(-3-1)} = -3*x^{-4}$
Evaluate at $x = 3$ : Evaluate the derivative at $x = 3$ . Now we substitute $x = 3$ into the derivative to find $g'(3)$ . $g'(3) = -3\cdot(3)^{-4}$
Simplify Expression: Simplify the expression. $g'(3) = -3\times\left(\frac{1}{3^4}\right) = -3\times\left(\frac{1}{81}\right) = -\frac{3}{81}$
Reduce Fraction: Reduce the fraction to its simplest form. $\newline$ $-\frac{3}{81}$ can be simplified by dividing both the numerator and the denominator by $3$ . $\newline$ $g'(3) = -\frac{1}{27}$