Final answer:
The slope of the tangent line to the graph of the function f(x) = 3x² + x at x=1 is found by differentiating the function, resulting in 6x + 1, and then plugging in x=1 to get a slope of 7.
Step-by-step explanation:
To find the slope of the tangent line to the graph of the function f(x) = 3x² + x at x=1, we need to calculate the derivative of the function, which gives us the slope of the tangent at any point x.
We can differentiate f(x) using the power rule for derivatives. For any term ax^n, the derivative is anx^(n-1). Applying this to f(x), we differentiate each term:
- The derivative of 3x² is 6x.
- The derivative of x is 1.
Thus, the derivative f'(x) is 6x + 1. To find the slope at x=1, we simply substitute 1 into the derivative:
f'(1) = 6(1) + 1 = 7
Therefore, the slope of the tangent line to the graph at x=1 is 7.