find the derivative of the function.\n\n$y = \\frac{5x - 9}{x^{2}+7x}$\n\n$y = \\square$
find the derivative of the function.\n\n$y = \\frac{5x - 9}{x^{2}+7x}$\n\n$y = \\square$
Answer
Explanation:
Step1: Apply quotient rule
Let ( u = 5x - 9), (u^\prime=5); (v=x^{2}+7x), (v^\prime = 2x + 7). The quotient rule is (y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}). So (y^\prime=\frac{5(x^{2}+7x)-(5x - 9)(2x + 7)}{(x^{2}+7x)^{2}}).
Step2: Expand numerator
Expand (5(x^{2}+7x)=5x^{2}+35x) and ((5x - 9)(2x + 7)=10x^{2}+35x-18x - 63=10x^{2}+17x - 63). Then (y^\prime=\frac{5x^{2}+35x-(10x^{2}+17x - 63)}{(x^{2}+7x)^{2}}).
Step3: Simplify numerator
(5x^{2}+35x - 10x^{2}-17x + 63=-5x^{2}+18x + 63). So (y^\prime=\frac{-5x^{2}+18x + 63}{(x^{2}+7x)^{2}}).
Answer:
(\frac{-5x^{2}+18x + 63}{(x^{2}+7x)^{2}})