evaluate (d)/(dx) ((x^(2)-cos x)) / (sin x+8x)

Question

Evaluate this question

evaluate (d)/(dx) ((x^(2)-cos x)) / (sin x+8x)

Answer 1

Let u=x²-cos(x), du/dx=2x+sin(x).

Let v=sin(x)+8x, dv/dx=cos(x)+8.

(d/dx)(u/v)=(vdu/dx-udv/dx)/v²=

((sin(x)+8x)(2x+sin(x))-(x²-cos(x))(cos(x)+8))/(sin(x)+8x)².

So (d/dx)((x²-cos(x))/(sin(x)+8x))=

((sin(x)+8x)(2x+sin(x))-(x²-cos(x))(cos(x)+8))/(sin(x)+8x)².

Expanding these terms:

(sin²(x)+10xsin(x)+16x²-(8x²+x²cos(x)-8cos(x)-cos²(x)))/(sin(x)+8x)²,

(sin²(x)+cos²(x)+8x²-x²cos(x)+10xsin(x)+8cos(x))/(sin(x)+8x)²,

(1+8x²-x²cos(x)+10xsin(x)+8cos(x))/(sin(x)+8x)².

So (d/dx)((x²-cos(x))/(sin(x)+8x))=

(1+8x²-x²cos(x)+10xsin(x)+8cos(x))/(sin(x)+8x)².