Final answer:
The slope of the tangent line to f(x) = ln(x³) - 9 at the point (1, -9) is 3. The equation of the tangent line is y = 3x - 12.
Step-by-step explanation:
Equation of the tangent line:
To find the equation of the tangent line to f(x) = ln(x³) - 9 at the point (1, -9), we need to find the slope of the tangent line first. The slope of the tangent line is equal to the derivative of the function evaluated at the given point. Taking the derivative of f(x) = ln(x³) - 9 using the chain rule, we get f'(x) = 3x²/x³ = 3/x. Evaluating the derivative at x = 1, we get f'(1) = 3/1 = 3. The slope of the tangent line is 3. Now we can use the point-slope form of the equation of a line y - y₁ = m(x - x₁), where (x₁, y₁) is the given point and m is the slope of the line. Plugging in the values, we get y - (-9) = 3(x - 1), which simplifies to y + 9 = 3x - 3. Rearranging the equation, we obtain the equation of the tangent line as y = 3x - 12.
Slope:
The slope of the tangent line to f(x) = ln(x³) - 9 at the point (1, -9) is 3.