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