Example Question - composition of functions

Finding Tangent Line to Composition of Functions

Given \( g(x) = \frac{16}{x} \) and \( f(x) = -4 \sqrt{x} \), find the tangent line to \( g(f(x)) \) at \( x=4 \). Step 1: Compute \( f(4) \). \( f(4) = -4 \sqrt{4} = -4 \cdot 2 = -8 \) Step 2: Compute \( g(f(4)) \). \( g(-8) = \frac{16}{-8} = -2 \) Step 3: Compute \( f'(x) \). \( f'(x) = -4 \cdot \frac{1}{2}x^{-\frac{1}{2}} = -2x^{-\frac{1}{2}} \) Step 4: Compute \( f'(4) \). \( f'(4) = -2 \cdot 4^{-\frac{1}{2}} = -2 \cdot \frac{1}{2} = -1 \) Step 5: Compute \( g'(x) \). \( g'(x) = -16x^{-2} \) Step 6: Compute \( g'(-8) \). \( g'(-8) = -16(-8)^{-2} = -16 \cdot \frac{1}{64} = -\frac{1}{4} \) Step 7: Apply chain rule to find derivative of \( g(f(x)) \) at \( x=4 \). \( (g(f(x)))' = g'(f(x)) \cdot f'(x) \) \( (g(f(4)))' = g'(-8) \cdot f'(4) = -\frac{1}{4} \cdot (-1) = \frac{1}{4} \) Step 8: Write the equation of the tangent line. \( y = mx + b \) \( m = \frac{1}{4} \) \( y = \frac{1}{4}x + b \) Step 9: Use point \( (f(4), g(f(4))) = (4, -2) \) to solve for \( b \). \( -2 = \frac{1}{4} \cdot 4 + b \) \( -2 = 1 + b \) \( b = -3 \) Step 10: Final equation of the tangent line. \( y = \frac{1}{4}x - 3 \)