Questions: Find the slope of the tangent line to the graph of the function at the given point. h(t)=t^2+6 t, (1,7) Use the limit process to find the derivative of the function. f(x)=5 f'(x)=

Solution Steps

To find the slope of the tangent line to the graph of the function \( h(t) = t^2 + 6t \) at the point (1, 7), we need to compute the derivative of \( h(t) \) and then evaluate it at \( t = 1 \).

To find the derivative of the constant function \( f(x) = 5 \) using the limit process, we recognize that the derivative of a constant function is always zero.

1. Compute the derivative of \( h(t) \).
2. Evaluate the derivative at \( t = 1 \).

1. Recognize that the derivative of a constant function is zero.

Given the function \( h(t) = t^2 + 6t \), we compute its derivative: \[ h'(t) = \frac{d}{dt}(t^2 + 6t) = 2t + 6 \]

To find the slope of the tangent line at the point \( (1, 7) \), we evaluate the derivative at \( t = 1 \): \[ h'(1) = 2(1) + 6 = 8 \]

For the constant function \( f(x) = 5 \), the derivative is: \[ f'(x) = 0 \]

Final Answer