y=((18x^(3)-5)/(3x^(3)))^(4)

Simplify Function: First, simplify the function before taking the derivative. y = ((18x^3 - 5) / (3x^3))^4 Simplify the fraction inside the parentheses by dividing both the numerator and the denominator by 3x^3. y = ((6x^3/3x^3) - (5/3x^3))^4 y = (2 - (5/3x^3))^4 Now the function is simplified.Apply Chain Rule: Next, apply the chain rule to find the derivative of y with respect to x. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. Let u = 2 - (5/3x^3), then y = u^4. First, find the derivative of y with respect to u, which is dy/du = 4u^3.Find u with respect to x: Now, find the derivative of u with respect to x, du/dx. u = 2 - (5/3x^3) du/dx = 0 - (-5 * 3x^(-4)) du/dx = 15x^(-4)Apply Chain Rule: Now, apply the chain rule by multiplying dy/du by du/dx to get dy/dx. dy/dx = dy/du * du/dx dy/dx = 4u^3 * 15x^(-4) Substitute u back into the equation. dy/dx = 4(2 - (5/3x^3))^3 * 15x^(-4)Simplify Derivative: Finally, simplify the derivative expression. dy/dx = 60x^(-4)(2 - (5/3x^3))^3 This is the derivative of the function y with respect to x.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

$y=\left(\frac{18 x^{3}-5}{3 x^{3}}\right)^{4}$

View step-by-step help

Home

Math Problems

Calculus

Find derivatives of using multiple formulae

Full solution

y=\left(\frac{18 x^{3}-5}{3 x^{3}}\right)^{4}

Simplify Function: First, simplify the function before taking the derivative. $\newline$ $y = \left(\frac{18x^3 - 5}{3x^3}\right)^4$ $\newline$ Simplify the fraction inside the parentheses by dividing both the numerator and the denominator by $3x^3$ . $\newline$ $y = \left(\frac{6x^3}{3x^3} - \frac{5}{3x^3}\right)^4$ $\newline$ $y = \left(2 - \frac{5}{3x^3}\right)^4$ $\newline$ Now the function is simplified.
Apply Chain Rule: Next, apply the chain rule to find the derivative of $y$ with respect to $x$ . The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. $\newline$ Let $u = 2 - \frac{5}{3}x^3$ , then $y = u^4$ . $\newline$ First, find the derivative of $y$ with respect to $u$ , which is $\frac{dy}{du} = 4u^3$ .
Find $u$ with respect to $x$ : Now, find the derivative of $u$ with respect to $x$ , $\frac{du}{dx}$ .
$u = 2 - \left(\frac{5}{3}x^3\right)$
$\frac{du}{dx} = 0 - \left(-5 \cdot 3x^{-4}\right)$
$\frac{du}{dx} = 15x^{-4}$
Apply Chain Rule: Now, apply the chain rule by multiplying $\frac{dy}{du}$ by $\frac{du}{dx}$ to get $\frac{dy}{dx}$ .
$\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$
$\frac{dy}{dx} = 4u^3 \times 15x^{-4}$
Substitute $u$ back into the equation.
$\frac{dy}{dx} = 4(2 - \frac{5}{3x^3})^3 \times 15x^{-4}$
Simplify Derivative: Finally, simplify the derivative expression. $\newline$ $\frac{dy}{dx} = 60x^{-4}(2 - \frac{5}{3x^3})^3$ $\newline$ This is the derivative of the function $y$ with respect to $x$ .