How do you estimate the gradient of y = x^2 + 3 at x = 1?

To estimate the gradient of \( y = x^2 + 3 \) at \( x = 1 \), use the derivative \( \frac{dy}{dx} = 2x \).

In more detail, the gradient of a curve at a specific point is given by the derivative of the function at that point. For the function \( y = x^2 + 3 \), we first need to find its derivative. The derivative of \( y = x^2 + 3 \) with respect to \( x \) is found by differentiating each term separately. The derivative of \( x^2 \) is \( 2x \), and the derivative of the constant \( 3 \) is \( 0 \). Therefore, the derivative of the function is \( \frac{dy}{dx} = 2x \).

To find the gradient at \( x = 1 \), we substitute \( x = 1 \) into the derivative. This gives us:

\[ \frac{dy}{dx} \bigg|_{x=1} = 2 \times 1 = 2 \]

So, the gradient of the curve \( y = x^2 + 3 \) at \( x = 1 \) is 2. This means that at the point where \( x = 1 \), the slope of the tangent to the curve is 2. This tells us how steep the curve is at that point.