f(x) = 4e^-1x +13x +5 at (0,9)

1. All steps for finding the first derivative of the function are clearly and accurately shown.
2. All steps for evaluating the derivative to find the slope of the tangent to the curve at the given point are clearly and accurately shown.
3. All steps for writing the equation of the tangent line to the curve at the given point are clearly and accurately shown.
4. The correct solution for EACH is shown.

what is the question?

1. To find the first derivative of the function f(x), we need to use the rules of differentiation. The first step is to identify each term in the function and apply the power rule and the chain rule as necessary.

The function is f(x) = 4e^(-1x) + 13x + 5.

Using the power rule, the derivative of e^(-1x) is (-1)e^(-1x) * (-1) = e^(-1x).

The derivative of 13x is 13.

Since the derivative of a constant term (5 in this case) is always 0, it does not affect the derivative.

So, the first derivative of f(x) is f'(x) = 4e^(-1x) + 13.

2. To find the slope of the tangent to the curve at the given point (0,9), we substitute x = 0 into the derivative we found in step 1.

Using f'(x) = 4e^(-1x) + 13, we evaluate at x = 0:

f'(0) = 4e^(-1(0)) + 13 = 4e^0 + 13 = 4(1) + 13 = 4 + 13 = 17.

So, the slope of the tangent to the curve at (0,9) is 17.

3. To write the equation of the tangent line to the curve at the given point, we need to use the point-slope form of a linear equation.

The point-slope form is given by: y - y1 = m(x - x1), where (x1, y1) is the point on the line and m is the slope.

We already have the slope (m) from step 2, which is 17, and the given point is (0,9).

Using the point-slope form, the equation of the tangent line is:
y - 9 = 17(x - 0).

Simplifying the equation, we get:
y - 9 = 17x.

4. The correct solutions for each step are:

1. f'(x) = 4e^(-1x) + 13.
2. f'(0) = 17.
3. The equation of the tangent line is y - 9 = 17x.