derivative of the function \( H(x)=\frac{\sqrt{x^{2}-3}}{\left(x^{2}+7\right)^{2}} \)

We wish to differentiate   H(x) = (√(x² − 3))⁄((x² + 7)²). Step 1. Write Numerator and Denominator.  Let   u(x) = √(x² − 3) = (x² − 3)^(1/2),   v(x) = (x² + 7)². Then, H(x) = u(x)/v(x). Step 2. Compute u′(x). Using the chain rule,   u′(x) = (1/2)(x² − 3)^(−1/2) · (2x) = x/√(x² − 3). Step 3. Compute v′(x). Since v(x) = (x² + 7)², we get   v′(x) = 2(x² + 7) · (2x) = 4x(x² + 7). Step 4. Apply the Quotient Rule. Recall the quotient rule:   H′(x) = [u′(x)v(x) − u(x)v′(x)] / [v(x)]². Plug in u, u′, v, and v′:   H′(x) = { [x/√(x² − 3)]·(x² + 7)² − √(x² − 3)·[4x(x² + 7)] } / [ (x² + 7)⁴ ]. Step 5. Simplify the Expression. Factor common terms in the numerator:  Both terms have an x and at least one factor of (x² + 7). Write:   Numerator = x(x² + 7) { (x² + 7)/√(x² − 3) − 4√(x² − 3) }. Thus,   H′(x) = x(x² + 7) { (x² + 7)/√(x² − 3) − 4√(x² − 3) } / (x² + 7)⁴ = x { (x² + 7)/√(x² − 3) − 4√(x² − 3) } / (x² + 7)³. We can combine the terms in curly brackets over a common denominator √(x² − 3):   (x² + 7)/√(x² − 3) − 4√(x² − 3) = [ (x² + 7) − 4(x² − 3) ] / √(x² − 3). Compute the numerator:   (x² + 7) − 4(x² − 3) = x² + 7 − 4x² + 12 = −3x² + 19. Thus, the derivative becomes:   H′(x) = [ x(−3x² + 19) ] / [ (x² + 7)³ √(x² − 3) ]. Step 6. Write the Final Answer. The derivative of the function H(x) is   H′(x) = x(19 − 3x²)⁄[(x² + 7)³ √(x² − 3)]  . (Notice that −3x² + 19 = 19 − 3x².)