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Given the function
y=(2x^(3)+1)/(3+x), find
(dy)/(dx) in simplified form.
Answer:
(dy)/(dx)=

Given the function $y=\frac{2 x^{3}+1}{3+x}$ , 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}+1}{3+x}

, 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 $y=\frac{2x^{3}+1}{3+x}$ , and 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, $u = 2x^3 + 1$ and $v = 3 + x$ .
Differentiate $u$ : Differentiate $u$ with respect to $x$ . The derivative of $u = 2x^3 + 1$ with respect to $x$ is $\frac{du}{dx} = 6x^2$ .
Differentiate $v$ : Differentiate $v$ with respect to $x$ . The derivative of $v = 3 + x$ with respect to $x$ is $\frac{dv}{dx} = 1$ .
Apply Quotient Rule: Apply the quotient rule using the derivatives from steps $3$ and $4$ . $\newline$ Substitute the derivatives into the quotient rule formula to get $\frac{dy}{dx} = \frac{(3 + x)(6x^2) - (2x^3 + 1)(1)}{(3 + x)^2}$ .
Simplify Expression: Simplify the expression. Expand the numerator to get $(dy)/(dx) = (18x^2 + 6x^2) - (2x^3 + 1)$ and simplify further.
Combine Like Terms: Combine like terms in the numerator. $\newline$ Combine the terms to get $(\frac{dy}{dx} = 24x^2 - 2x^3 - 1)$ .
Factor Numerator: Factor the numerator if possible. $\newline$ In this case, the numerator cannot be factored in a way that will cancel with the denominator, so we leave the expression as is.
Final Simplified Form: Write the final simplified form of the derivative. $\newline$ The final simplified form of the derivative is $(dy)/(dx) = (24x^2 - 2x^3 - 1)/((3 + x)^2)$ .

More problems from Find trigonometric ratios using reference angles

Question

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\newline

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