differentiate y=(x^3-7x^2)(1+9x)^1/2.

Question

Differentiate

Answer 1

Let u=x³-7x² and v=√(1+9x²), then:

du/dx=3x²-14x and dv/dx=9x/√(1+9x²).

dy/dx=v(du/dx)+u(dv/dx)=(3x²-14x)√(1+9x²)+9x(x³-7x²)/√(1+9x²).

This can be rationalised partly:

(3x²-14x)√(1+9x²)+9x(x³-7x²)√(1+9x²)/(1+9x²),

and then simplified:

[(3x²-14x)(1+9x²)+9x(x³-7x²)]√(1+9x²)/(1+9x²),

(3x²+27x⁴-14x-126x³+9x⁴-63x³)√(1+9x²)/(1+9x²),

x(36x³-189x²+3x-14)√(1+9x²)/(1+9x²).