Given the function f(x)=(x)/(1-x^(3)), find f^(')(x) in simplified form. Answer: f^(')(x)=

Identify Function: Identify the function to differentiate. We are given the function f(x) = (x)/(1-x^3). We need to find its derivative, which is denoted by f^(')(x).Apply Quotient Rule: Apply the quotient rule for differentiation. The quotient rule states that if we have a function g(x) = u(x)/v(x), then g^(')(x) = (u^(')(x)v(x) - u(x)v^(')(x))/(v(x))^2. Here, u(x) = x and v(x) = 1 - x^3.Differentiate u and v: Differentiate u(x) and v(x). The derivative of u(x) = x with respect to x is u^(')(x) = 1. The derivative of v(x) = 1 - x^3 with respect to x is v^(')(x) = -3x^2.Apply Derivatives: Apply the derivatives to the quotient rule. Using the derivatives from Step 3, we substitute into the quotient rule formula: f^(')(x) = (1 * (1 - x^3) - x * (-3x^2)) / (1 - x^3)^2.Simplify Numerator: Simplify the numerator. Simplify the expression in the numerator: f^(')(x) = (1 - x^3 + 3x^3) / (1 - x^3)^2.Combine Like Terms: Combine like terms in the numerator. Combine the x^3 terms: f^(')(x) = (1 + 2x^3) / (1 - x^3)^2.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Given the function
f(x)=(x)/(1-x^(3)), find
f^(')(x) in simplified form.
Answer:
f^(')(x)=

Given the function $f(x)=\frac{x}{1-x^{3}}$ , find $f^{\prime}(x)$ in simplified form. $\newline$ Answer: $f^{\prime}(x)=$

View step-by-step help

Home

Math Problems

Algebra 2

Evaluate exponential functions

Full solution

Q. Given the function

f(x)=\frac{x}{1-x^{3}}

, find

f^{\prime}(x)

in simplified form.

\newline

Answer:

f^{\prime}(x)=

Identify Function: Identify the function to differentiate. $\newline$ We are given the function $f(x) = \frac{x}{1-x^3}$ . We need to find its derivative, which is denoted by $f^{\prime}(x)$ .
Apply Quotient Rule: Apply the quotient rule for differentiation. The quotient rule states that if we have a function $g(x) = \frac{u(x)}{v(x)}$ , then $g^{\prime}(x) = \frac{u^{\prime}(x)v(x) - u(x)v^{\prime}(x)}{(v(x))^2}$ . Here, $u(x) = x$ and $v(x) = 1 - x^3$ .
Differentiate $u$ and $v$ : Differentiate $u(x)$ and $v(x)$ . The derivative of $u(x) = x$ with respect to $x$ is $u^{'}(x) = 1$ . The derivative of $v(x) = 1 - x^3$ with respect to $x$ is $v^{'}(x) = -3x^2$ .
Apply Derivatives: Apply the derivatives to the quotient rule. $\newline$ Using the derivatives from Step $3$ , we substitute into the quotient rule formula: $\newline$ $f'(x) = \frac{1 \cdot (1 - x^3) - x \cdot (-3x^2)}{(1 - x^3)^2}$ .
Simplify Numerator: Simplify the numerator. $\newline$ Simplify the expression in the numerator: $\newline$ $f^{\prime}(x) = \frac{1 - x^3 + 3x^3}{(1 - x^3)^2}$ .
Combine Like Terms: Combine like terms in the numerator. $\newline$ Combine the $x^3$ terms: $\newline$ $f^{\prime}(x) = \frac{1 + 2x^3}{(1 - x^3)^2}.$