Compute the derivative of the given function and find the slope of the line that is tangent to its graph for the specified value f the independent variable. f(x) = 9x^2 – 2x + 8; x =3.
To compute the derivative of the given function, we can use the power rule of differentiation. The power rule states that if we have a term of the form x^n, then the derivative is given by nx^(n-1).
In this case, the function f(x) = 9x^2 – 2x + 8. We'll differentiate each term separately:
- The derivative of 9x^2 is (2)(9)x^(2-1) = 18x.
- The derivative of -2x is (1)(-2)x^(1-1) = -2.
- The derivative of 8 is 0, since it's a constant.
Now, we can add up the derivatives of each term to get the derivative of the entire function:
f'(x) = 18x - 2 + 0
We have the derivative, and now we need to find the slope of the line that is tangent to the graph of the function at x = 3. We can plug in x = 3 into the derivative, f'(x), and evaluate it to find the slope:
f'(3) = 18(3) - 2
f'(3) = 54 - 2
f'(3) = 52
Therefore, the slope of the line that is tangent to the graph of f(x) = 9x^2 – 2x + 8 at x = 3 is 52.