differentiate. g(x)=\frac{5x - 3}{8x + 5}+x^{3} g(x)=square

Explanation:

Step1: Apply sum - rule of differentiation

The sum - rule states that if (y = u + v), then (y'=u'+v'). Let (u=\frac{5x - 3}{8x + 5}) and (v = x^{3}). So (g'(x)=u'+v').

Step2: Differentiate (v = x^{3})

Using the power - rule ((x^{n})'=nx^{n - 1}), for (v=x^{3}), we have (v' = 3x^{2}).

Step3: Differentiate (u=\frac{5x - 3}{8x + 5}) using the quotient - rule

The quotient - rule states that if (u=\frac{f(x)}{g(x)}), then (u'=\frac{f'(x)g(x)-f(x)g'(x)}{g(x)^{2}}). Here, (f(x)=5x - 3), (f'(x)=5), (g(x)=8x + 5), (g'(x)=8). Then (u'=\frac{5(8x + 5)-(5x - 3)\times8}{(8x + 5)^{2}}=\frac{40x+25-(40x - 24)}{(8x + 5)^{2}}=\frac{40x + 25-40x + 24}{(8x + 5)^{2}}=\frac{49}{(8x + 5)^{2}}).

Step4: Find (g'(x))

Since (g'(x)=u'+v'), substituting (u') and (v') we get (g'(x)=\frac{49}{(8x + 5)^{2}}+3x^{2}).

Answer:

(\frac{49}{(8x + 5)^{2}}+3x^{2})