Step 1: Understanding the Question:
The slope of a tangent line to a curve at a given point is physically and geometrically represented by the value of the first derivative of the curve's equation at that point.
Therefore, the task is to differentiate the polynomial function \( y \) and evaluate the resulting expression at the specific input \( x = 1 \). Step 2: Key Formula or Approach:
1. Derivative of a polynomial: \( \frac{d}{dx}(x^n) = nx^{n-1} \).
2. Slope of tangent \( m = \left. \frac{dy}{dx} \right|_{x=x_0} \). Step 3: Detailed Explanation:
The given curve equation is \( y = 9x^2 + 7x^4 + 5 \).
Differentiate each term using the power rule:
\[ \frac{dy}{dx} = \frac{d}{dx}(9x^2) + \frac{d}{dx}(7x^4) + \frac{d}{dx}(5) \]
\[ \frac{dy}{dx} = 9(2x) + 7(4x^3) + 0 \]
\[ \frac{dy}{dx} = 18x + 28x^3 \]
Now, we evaluate this derivative at the point where \( x = 1 \):
\[ m = 18(1) + 28(1)^3 \]
\[ m = 18 + 28(1) \]
\[ m = 18 + 28 \]
\[ m = 46 \]
Thus, the instantaneous rate of change of the function at \( x=1 \), which is the slope of the tangent line, is 46.
Step 4: Final Answer:
The slope of the tangent is 46.