Derivatives of functions

Question

Find the derivative of the function.
h(x) = x^2 arctan(7x)

Answer 1

Hi Ketevan,

 

The derivative for arctan(x) = 1 / (x^2 + 1).

 

We need to use the product rule here. Remember that the product rule is the derivative of f(x)*g(x) = f'(x)g(x) + f(x)g'(x).

 

We also use the chain rule. Remember that the derivative of f(g(x)) = f'(g(x)) * g'(x) * x' (but x' is 1).

 

Using those two, we get:

 

h'(x) = 2x * arctan(7x) + (x^2) / (49x^2 + 1) * 7

 

To simplify,

 

h'(x) = 2xarctan(7x) + (7x^2)/(49x^2+1)

 

Hope this helps!

Answer 2

Use product rule and trigonometric differentiation.

 

h'(x) = 2xarctan(7x) + (x² * 7 / 1 + 49x²)

 

h'(x) = 2xarctan(7x) + [7x² / (1 + 49x²)]

 

 

 

Note that  dy/dx[arctan(x)] = 1 / (1 + x²)