Final answer:
The derivative of f(x) = 7x^2 - 3x is 14x - 3. The slope of the tangent line at x = 4 is 53. The equation of the tangent line to the graph of f at x = 4 is y = 53x - 112.
Step-by-step explanation:
To find the derivative of f(x) = 7x^2 - 3x, we can use the power rule of differentiation. The power rule states that if we have a function f(x) = cx^n, where c is a constant and n is a real number, then the derivative of f(x) is given by f'(x) = cnx^(n-1).
Step 1:
Apply the power rule to find the derivative of 7x^2 - 3x.
f'(x) = 2(7)x^(2-1) - 1(3)x^(1-1)
= 14x - 3
Step 2:
Plug in the value x = 4 to find the slope at x = 4.
f'(4) = 14(4) - 3
= 56 - 3
= 53
Therefore, the slope of the tangent line at x = 4 is 53.
Using the point-slope form of a linear equation, we can find the equation of the tangent line.
Step 3:
Find the point on the graph of f(x) at x = 4.
f(4) = 7(4)^2 - 3(4)
= 112 - 12
= 100
Step 4:
Plug in the values m = 53 (slope) and (x, y) = (4, 100) (point on the line) into the point-slope form of a linear equation: y - y1 = m(x - x1).
y - 100 = 53(x - 4)
y - 100 = 53x - 212
y = 53x - 112
Therefore, the equation of the tangent line to the graph of f at x = 4 is y = 53x - 112.