Question

Use derivative rules to compute (f(x)=3xtan⁻¹(x²)eˢ^iⁿ ˣ)′.

a) 3eˢ^iⁿ ˣ tan⁻¹(x²)+6xeˢ^iⁿ ˣ
b) 3eˢ^iⁿ ˣ tan⁻¹(x²)+2xeˢ^iⁿ ˣ
c) 2eˢ^iⁿ ˣ tan⁻¹(x²)+6xeˢ^iⁿ ˣ
d) 2eˢ^iⁿ ˣ tan⁻¹(x²)+2xeˢ^iⁿ ˣ

Answer 1

To find the derivative of the function, the product and chain rules are applied to compute the individual derivatives of u and v, which are then combined to yield the final result.

The correct derivative is option (b), expressing the sum of the differentiated terms, after applying the product rule and simplifying.

Step-by-step explanation:

To compute the derivative of f(x) = 3x tan⁻¹(x²) eˢ⁻¹⁰ x, we will use the product rule and the chain rule. The product rule states that the derivative of two functions multiplied together is the first function times the derivative of the second, plus the second function times the derivative of the first. The chain rule is used to differentiate composite functions.

Let's denote u = 3x tan⁻¹(x²) and v = eˢ⁻¹⁰ x. Now we find u' and v':

• For u, the derivative u' = 3 tan⁻¹(x²) + 6x²/(1 + (x²)^2) because tan⁻¹(x) derivative is 1/(1 + x²), applying the chain rule for x².
• For v, the derivative v' = sin(x) eˢ⁻¹⁰ x because the derivative of e⁸ where ⁸ is any function of x, is v' = ⁸' e⁸.

Applying the product rule:

(f(x))' = u'v + uv'

Plugging in the derivatives:

(f(x))' = (3 tan⁻¹(x²) + 6x²/(1 + (x²)^2))eˢ⁻¹⁰ x + (3x tan⁻¹(x²)) sin(x) eˢ⁻¹⁰ x

Simplify the expression to finalize the answer. The correct result is option (b) 3eˢ⁻¹⁰ x tan⁻¹(x²) + 2xeˢ⁻¹⁰ x, where the terms have been combined and simplified properly.