Macdonald Osborne
11/27/2023 · Middle School
Differentiate \( g(x)=\frac{5 x-1}{8 x+9}+x^{3} \)
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Tutor-Verified Answer
Step-by-step Solution
Find the first order derivative with respect to \( x \) for \( \frac{5x-1}{8x+9}+x^3 \).
Evaluate the derivative by following steps:
- step0: Evaluate the derivative:
\(\frac{d}{dx}\left(\frac{5x-1}{8x+9}+x^{3}\right)\)
- step1: Add the terms:
\(\frac{d}{dx}\left(\frac{5x-1+8x^{4}+9x^{3}}{8x+9}\right)\)
- step2: Use differentiation rules:
\(\frac{\frac{d}{dx}\left(5x-1+8x^{4}+9x^{3}\right)\times \left(8x+9\right)-\left(5x-1+8x^{4}+9x^{3}\right)\times \frac{d}{dx}\left(8x+9\right)}{\left(8x+9\right)^{2}}\)
- step3: Calculate:
\(\frac{\left(5+32x^{3}+27x^{2}\right)\left(8x+9\right)-\left(5x-1+8x^{4}+9x^{3}\right)\times 8}{\left(8x+9\right)^{2}}\)
- step4: Calculate:
\(\frac{\left(5+32x^{3}+27x^{2}\right)\left(8x+9\right)-8\left(5x-1+8x^{4}+9x^{3}\right)}{\left(8x+9\right)^{2}}\)
- step5: Calculate:
\(\frac{53+192x^{4}+432x^{3}+243x^{2}}{\left(8x+9\right)^{2}}\)
- step6: Calculate:
\(\frac{53+192x^{4}+432x^{3}+243x^{2}}{64x^{2}+144x+81}\)
The first derivative of the function \( g(x)=\frac{5x-1}{8x+9}+x^{3} \) is:
\[ g'(x) = \frac{53+192x^{4}+432x^{3}+243x^{2}}{64x^{2}+144x+81} \]
Quick Answer
\[ g'(x) = \frac{53+192x^{4}+432x^{3}+243x^{2}}{64x^{2}+144x+81} \]
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