Given the function $f\left(x\right)=x^2+3x$, what is the slope of the line tangent to the curve at the point $x=2$?

• A. $4$
• B. $9$
• C. $5$
• D. $7$

Answer: (D)

Verified step by step guidance

1. Step 1: Recall that the slope of the tangent line to a curve at a given point is determined by the derivative of the function at that point. The derivative of a function, f'(x), represents the rate of change of the function with respect to x.
2. Step 2: Start by finding the derivative of the given function f(x) = x^2 + 3x. Use the power rule for differentiation, which states that the derivative of x^n is n*x^(n-1), and the constant multiple rule, which states that the derivative of c*x is c.
3. Step 3: Apply the power rule to x^2, which gives 2x, and apply the constant multiple rule to 3x, which gives 3. Combine these results to find the derivative: f'(x) = 2x + 3.
4. Step 4: Evaluate the derivative at the given point x = 2. Substitute x = 2 into f'(x) = 2x + 3 to find the slope of the tangent line at that point.
5. Step 5: Simplify the expression obtained in Step 4 to determine the slope of the tangent line. The result will be the slope of the line tangent to the curve at x = 2.