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Differentiate w.r.t. x
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\frac{5}{\left(x-8\right)\left(x-1\right)}+\frac{3}{\left(x-8\right)\left(x+2\right)}
Factor x^{2}-9x+8. Factor x^{2}-6x-16.
\frac{5\left(x+2\right)}{\left(x-8\right)\left(x-1\right)\left(x+2\right)}+\frac{3\left(x-1\right)}{\left(x-8\right)\left(x-1\right)\left(x+2\right)}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-8\right)\left(x-1\right) and \left(x-8\right)\left(x+2\right) is \left(x-8\right)\left(x-1\right)\left(x+2\right). Multiply \frac{5}{\left(x-8\right)\left(x-1\right)} times \frac{x+2}{x+2}. Multiply \frac{3}{\left(x-8\right)\left(x+2\right)} times \frac{x-1}{x-1}.
\frac{5\left(x+2\right)+3\left(x-1\right)}{\left(x-8\right)\left(x-1\right)\left(x+2\right)}
Since \frac{5\left(x+2\right)}{\left(x-8\right)\left(x-1\right)\left(x+2\right)} and \frac{3\left(x-1\right)}{\left(x-8\right)\left(x-1\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.
\frac{5x+10+3x-3}{\left(x-8\right)\left(x-1\right)\left(x+2\right)}
Do the multiplications in 5\left(x+2\right)+3\left(x-1\right).
\frac{8x+7}{\left(x-8\right)\left(x-1\right)\left(x+2\right)}
Combine like terms in 5x+10+3x-3.
\frac{8x+7}{x^{3}-7x^{2}-10x+16}
Expand \left(x-8\right)\left(x-1\right)\left(x+2\right).