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Given the function
y=(2x^(3)-3)/(3+2x^(3)), find
(dy)/(dx) in simplified form.
Answer:
(dy)/(dx)=

Given the function $y=\frac{2 x^{3}-3}{3+2 x^{3}}$ , find $\frac{d y}{d x}$ in simplified form. $\newline$ Answer: $\frac{d y}{d x}=$

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Find trigonometric ratios using reference angles

Full solution

Q. Given the function

y=\frac{2 x^{3}-3}{3+2 x^{3}}

, find

\frac{d y}{d x}

in simplified form.

\newline

Answer:

\frac{d y}{d x}=

Identify Function: Identify the function that needs to be differentiated. $\newline$ We are given the function $y=\frac{2x^{3}-3}{3+2x^{3}}$ . We need to find its derivative with respect to $x$ , which is denoted as $\frac{dy}{dx}$ .
Apply Quotient Rule: Apply the quotient rule for differentiation. $\newline$ The quotient rule states that if we have a function $y = \frac{u}{v}$ , where both $u$ and $v$ are functions of $x$ , then the derivative of $y$ with respect to $x$ is given by $\frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$ . Here, $u = 2x^3 - 3$ and $v = 3 + 2x^3$ .
Differentiate $u$ : Differentiate $u$ with respect to $x$ . The derivative of $u = 2x^3 - 3$ with respect to $x$ is $\frac{du}{dx} = 6x^2$ .
Differentiate $v$ : Differentiate $v$ with respect to $x$ . The derivative of $v = 3 + 2x^3$ with respect to $x$ is $\frac{dv}{dx} = 6x^2$ .
Apply Quotient Rule: Apply the quotient rule using the derivatives from steps $3$ and $4$ . $\newline$ Substitute $(du)/(dx)$ and $(dv)/(dx)$ into the quotient rule formula to get $(dy)/(dx) = \frac{(3 + 2x^3)(6x^2) - (2x^3 - 3)(6x^2)}{(3 + 2x^3)^2}$ .
Expand Numerator: Expand the numerator of the derivative. $\newline$ Expanding the numerator, we get $(dy)/(dx) = (18x^2 + 12x^5 - 12x^5 + 18x^2)/((3 + 2x^3)^2)$ .
Simplify Numerator: Simplify the numerator. $\newline$ Notice that the terms $+12x^5$ and $-12x^5$ cancel each other out. So, we have $(dy)/(dx) = (18x^2 + 18x^2)/((3 + 2x^3)^2)$ .
Combine Like Terms: Combine like terms in the numerator. Combining like terms, we get $\frac{dy}{dx} = \frac{36x^2}{(3 + 2x^3)^2}$ .

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