Differentiate the function \( G(x)=-7 \sqrt{x} \sec x \)

To differentiate the function \( G(x) = -7 \sqrt{x} \sec x \), we apply the product rule of differentiation. The product rule states that if you have a function that is the product of two functions, say \( u(x) \) and \( v(x) \), then the derivative is given by \( (uv)' = u'v + uv' \). Here, let \( u(x) = -7 \sqrt{x} \) and \( v(x) = \sec x \). 1. The derivative of \( u(x) \): \[ u'(x) = -7 \cdot \frac{1}{2\sqrt{x}} = -\frac{7}{2\sqrt{x}} \] 2. The derivative of \( v(x) \): \[ v'(x) = \sec x \tan x \] Now applying the product rule: \[ G'(x) = u'v + uv' = \left(-\frac{7}{2\sqrt{x}}\right)(\sec x) + \left(-7 \sqrt{x}\right)(\sec x \tan x) \] This simplifies to: \[ G'(x) = -\frac{7\sec x}{2\sqrt{x}} - 7\sqrt{x} \sec x \tan x \] Thus, the derivative of \( G(x) \) is: \[ G'(x) = -\frac{7 \sec x}{2\sqrt{x}} - 7 \sqrt{x} \sec x \tan x \]