Solve for the derivative of y = ((6x2+1)3 * (2x−5)4) / ((x3-2x+1)2)

Question

ScanSolve AI Tutor · Accepted Answer

The problem asks to "Solve this", which for a complex algebraic expression typically means finding its derivative. We will use logarithmic differentiation to find the derivative of the given function.

Let $y$ be the given expression:

y = \frac{(6x^2+1)^3 (2x-5)^4}{(x^3-2x+1)^2}

Take the natural logarithm of both sides.
$\ln y = \ln \left( \frac{(6x^2+1)^3 (2x-5)^4}{(x^3-2x+1)^2} \right)$
Use logarithm properties ( $\ln(AB) = \ln A + \ln B$ , $\ln(\frac{A}{B}) = \ln A - \ln B$ , $\ln(A^n) = n \ln A$ ) to expand the right side.
$\ln y = 3 \ln(6x^2+1) + 4 \ln(2x-5) - 2 \ln(x^3-2x+1)$
Differentiate both sides with respect to $x$ .
Remember that $\frac{d}{dx}(\ln u) = \frac{1}{u} \frac{du}{dx}$ .
$\frac{1}{y} \frac{dy}{dx} = 3 \cdot \frac{1}{6x^2+1} \cdot \frac{d}{dx}(6x^2+1) + 4 \cdot \frac{1}{2x-5} \cdot \frac{d}{dx}(2x-5) - 2 \cdot \frac{1}{x^3-2x+1} \cdot \frac{d}{dx}(x^3-2x+1)$
Calculate the derivatives of the inner functions and substitute them back.
$\frac{d}{dx}(6x^2+1) = 12x$ $\frac{d}{dx}(2x-5) = 2$ $\frac{d}{dx}(x^3-2x+1) = 3x^2-2$
Substituting these into the equation from Step 3:
$\frac{1}{y} \frac{dy}{dx} = 3 \cdot \frac{12x}{6x^2+1} + 4 \cdot \frac{2}{2x-5} - 2 \cdot \frac{3x^2-2}{x^3-2x+1}$ $\frac{1}{y} \frac{dy}{dx} = \frac{36x}{6x^2+1} + \frac{8}{2x-5} - \frac{2(3x^2-2)}{x^3-2x+1}$
Multiply both sides by $y$ to solve for $\frac{dy}{dx}$ .
$\frac{dy}{dx} = y \left( \frac{36x}{6x^2+1} + \frac{8}{2x-5} - \frac{2(3x^2-2)}{x^3-2x+1} \right)$
Substitute the original expression for $y$ back into the equation.
$\frac{dy}{dx} = \frac{(6x^2+1)^3 (2x-5)^4}{(x^3-2x+1)^2} \left( \frac{36x}{6x^2+1} + \frac{8}{2x-5} - \frac{2(3x^2-2)}{x^3-2x+1} \right)$
The final answer is:
$\boxed{\frac{dy}{dx} = \frac{(6x^2+1)^3 (2x-5)^4}{(x^3-2x+1)^2} \left( \frac{36x}{6x^2+1} + \frac{8}{2x-5} - \frac{2(3x^2-2)}{x^3-2x+1} \right)}$

Point	x	y
Minimum	−0.06	584.21
y-intercept	0	625

Solve for the derivative of y = ((6x2+1)3 * (2x-5)4) / ((x3-2x+1)2)

Step-by-step solution

Take the natural logarithm of both sides.

Use logarithm properties ( $\ln(AB) = \ln A + \ln B$ , $\ln(\frac{A}{B}) = \ln A - \ln B$ , $\ln(A^n) = n \ln A$ ) to expand the right side.

Differentiate both sides with respect to $x$ .

Calculate the derivatives of the inner functions and substitute them back.

Multiply both sides by $y$ to solve for $\frac{dy}{dx}$ .

Substitute the original expression for $y$ back into the equation.

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