First of all, let's understand what is being asked. Our task is to find the equation of the tangent line to the given function at the point (0,-5). This is a common task in calculus and requires us to understand a couple of things - what a tangent line is, how to calculate it, and then how to formulate it as an equation.
So what is a tangent line? A tangent line to the function at a certain point is a straight line that just "touches" the function at that point. In other words, it's a line that has the same slope as the function at that point.
The slope of a function at any point is given by the derivative of that function at that point. The derivative of a function can be understood as a function itself, which gives the slopes of the original function. Therefore, to calculate the slope of the tangent line, we will first need to calculate the derivative of the function.
Let's start with the given function f(x) = -5x - 9 + 4e^x. The derivative of a function can be computed term-by-term, and in our case, it's also pretty straightforward:
The derivative of -5x is -5, the derivative of -9 is 0 (since it's a constant), and the derivative of 4e^x is 4e^x, because the derivative of e^x is e^x itself. Combining these, we get that the derivative f'(x) of our function is -5 + 4e^x.
We need the derivative at the point x=0, so we'll substitute this into our derivative: f'(0) = -5 + 4e^0 = -5 + 4*1, which simplifies to -1.
So, the slope of the tangent line at the point (0,-5) is -1.
The last step is to find the equation of the tangent line. We can use the point-slope form of the linear equation, which is y - y1 = m * (x - x1), where m is the slope of the line, and (x1, y1) is the point that it goes through. We substitute m = -1, x1 = 0 and y1 = -5 to get:
y - (-5) = -1 * (x - 0)
which simplifies to y + 5 = -x
Finally, we get the equation of the tangent line: y = -x - 5.