Find the derivative of the function.
$f\left(x\right)=4x^2\sec x-\sqrt{x}$

• A. $8x\sec x+\frac{1}{2\sqrt{x}}$
• B. $8x\sec x\tan x-\frac{1}{2\sqrt{x}}$
• C. $4x^2\sec x\tan x+8x\sec x-\frac12\sqrt{x}$
• D. $4x^2\sec x\tan x+8x\sec x-\frac{1}{2\sqrt{x}}$

Answer: (D)

Verified step by step guidance

1. Step 1: Identify the function to differentiate. The given function is f(x) = 4x^2 sec(x) - sqrt(x). This function consists of two terms: 4x^2 sec(x) and -sqrt(x). We will differentiate each term separately using the rules of differentiation.
2. Step 2: Differentiate the first term, 4x^2 sec(x). Use the product rule since this term is a product of 4x^2 and sec(x). The product rule states that if u(x) and v(x) are functions, then d/dx[u(x)v(x)] = u'(x)v(x) + u(x)v'(x). Here, u(x) = 4x^2 and v(x) = sec(x). Differentiate u(x) to get u'(x) = 8x, and differentiate v(x) to get v'(x) = sec(x)tan(x). Substitute these into the product rule formula.
3. Step 3: Differentiate the second term, -sqrt(x). Recall that sqrt(x) can be written as x^(1/2). Use the power rule for differentiation, which states that d/dx[x^n] = n*x^(n-1). Here, n = 1/2, so the derivative of -sqrt(x) is -1/2 * x^(-1/2). Simplify this to -1/(2sqrt(x)).
4. Step 4: Combine the results from Steps 2 and 3. The derivative of the first term, 4x^2 sec(x), is 8x sec(x) + 4x^2 sec(x)tan(x). The derivative of the second term, -sqrt(x), is -1/(2sqrt(x)). Add these together to get the derivative of the entire function.
5. Step 5: Write the final expression for the derivative. Combine all terms to get f'(x) = 4x^2 sec(x)tan(x) + 8x sec(x) - 1/(2sqrt(x)). This is the derivative of the given function.