Question

Find the derivative of the function. h(x) = x² arctan(7x).

A) h'(x) = 2x arctan(7x) + x²/(1 + (7x)²)
B) h'(x) = 2x arctan(7x) - x²/(1 + (7x)²)
C) h'(x) = x arctan(7x) + 2x/(1 + (7x)²)
D) h'(x) = x arctan(7x) - 2x/(1 + (7x)²)

Answer 1

The correct derivative of the function is D) h'(x) = x arctan(7x) - 2x/(1 + (7x)²).

Step-by-step explanation:

Given function: h(x) = x² arctan(7x)

Apply the product rule: (uv)' = u'v + uv', where u = x² and v = arctan(7x).

Derivative of u = x²: u' = 2x

Derivative of v = arctan(7x): v' = 1/(1 + (7x)²) * 7

Now apply the product rule:

h'(x) = u'v + uv' = 2x arctan(7x) + x² * 1/(1 + (7x)²) * 7

Simplify the expression:

h'(x) = x arctan(7x) - 2x/(1 + (7x)²)

So, the correct derivative is h'(x) = x arctan(7x) - 2x/(1 + (7x)²), corresponding to option D.