Question

For some constants a and b, find the derivative of (ax² + b)².

Answer 1

Let f(x) = (ax² + b²)

= f(x) = a²x⁴ + 2abx² + b²

∴ `f'(x) = d/(dx) (a^2 x^4 + 2abx^2 + b^2) = a^2 d/(dx) (x^4) + 2ab d/(dx) (x^2) + d/(dx)(b^2)`

Using the theorem `d/(dx) x^n = nx^(n - 1)`, we obtain

∴ f'(x) = a² (4x³) + 2ab (2x) + b²0

= 4a²x³ + 4abx

= 4ax(ax² + b)