Question

Compute the derivative for each of the following functions:

(a) f(x) = cos(3x) + sin(x² + x)
(b) g(x) = sqrt(1 + tan(x))
(c) h(x) = ln(1 - cos(2x))
(d) i(x) = x*arctan(2x)

Answer 1

To compute the derivatives for the given functions, we use the chain rule, product rule, and basic derivative rules. (a) f'(x) = -3sin(3x) + (2x + 1)cos(x² + x). (b) g'(x) = sec²(x) / (2sqrt(1 + tan(x))). (c) h'(x) = -2sin(2x) / (1 - cos(2x)). (d) i'(x) = arctan(2x) + x * (1 / (1 + (2x)²)).

Step-by-step explanation:

(a) To compute the derivative of the function f(x) = cos(3x) + sin(x² + x), we need to apply the chain rule and product rule. Taking the derivative of each term individually and adding them together, we get:

f'(x) = -3sin(3x) + (2x + 1)cos(x² + x)

(b) The derivative of the function g(x) = sqrt(1 + tan(x)) can be found using the chain rule. Applying the chain rule, we get:

g'(x) = sec²(x) / (2sqrt(1 + tan(x)))

(c) To find the derivative of h(x) = ln(1 - cos(2x)), we apply the chain rule. Using the chain rule, we have:

h'(x) = -2sin(2x) / (1 - cos(2x))

(d) The derivative of i(x) = x*arctan(2x) can be found using the product rule. Applying the product rule, we get:

i'(x) = arctan(2x) + x * (1 / (1 + (2x)²))