Final answer:
The slope of the tangent line to the graph of the function f(x) = x⁻4 + 25x² + 144 at a given value of x can be found by taking the derivative of the function f(x) with respect to x and then evaluating it at that specific x-value.
Step-by-step explanation:
To find the slope of the tangent line at a particular point on the graph of the function, we first need to determine the derivative of the function. The given function f(x) = x⁻4 + 25x² + 144 can be rewritten as f(x) = x⁻4 + 25x² + 144x⁰. Taking the derivative of f(x) using the power rule, we get f'(x) = -4x⁻5 + 50x.
Now, to find the slope of the tangent line at a specific x-value, say x = a, substitute this value into the derivative f'(x). Plugging a into f'(x) gives f'(a) = -4a⁻5 + 50a. This value represents the slope of the tangent line to the graph of f(x) at the point where x = a.
The equation of the tangent line at this specific x-value (x = a) can be determined using the point-slope form: y - y₁ = m(x - x₁), where (x₁, y₁) represents the point on the graph and m is the slope of the tangent line. Substituting the x-value a and the corresponding y-value obtained by evaluating f(a) into the equation will give the equation of the tangent line at x = a.