Questions: Find the slope of the tangent line to the graph of the function at the given point. f(x)=3 x^2-6, (2,6)

Solution Steps

To find the slope of the tangent line to the graph of the function at a given point, we need to compute the derivative of the function and then evaluate it at the given x-coordinate of the point. The derivative of a function gives us the slope of the tangent line at any point on the function.

The function given is \( f(x) = 3x^2 - 6 \).

To find the slope of the tangent line, we first need to compute the derivative of the function. The derivative of \( f(x) = 3x^2 - 6 \) with respect to \( x \) is: \[ f'(x) = \frac{d}{dx}(3x^2 - 6) = 6x \]

We need to find the slope of the tangent line at the point \( (2, 6) \). To do this, we evaluate the derivative at \( x = 2 \): \[ f'(2) = 6 \times 2 = 12 \]

Final Answer