Question

Let f(x) = e ^3x/5x − 2. Find f'(0).

Answer 1

Answer:

Step-by-step explanation:

Our friend asking what the actual function is has a point. I completed this under the assumption that what we have is:

$f(x)=\frac{e^{3x}}{5x-2}$ and used the quotient rule to find the derivative, as follows:

$f'(x)=\frac{e^{3x}(5)-[(5x-2)(3e^{3x})]}{(5x-2)^2}$ and simplifying a bit:

$f'(x)=\frac{5e^{3x}-[15xe^{3x}-6e^{3x}]}{(5x-2)^2}$and a bit more to:

$f'(x)=\frac{5e^{3x}-15xe^{3x}+6e^{3x}}{(5x-2)^2}$ and combining like terms:

$f'(x)=\frac{11e^{3x}-15xe^{3x}}{(5x-2)^2}$ and factor out the GFC in the numerator to get:

$f'(x)=\frac{e^{3x}(11-15x)}{(5x-2)^2}$  That's the derivative simplified. If we want f'(0), we sub in 0's for the x's in there and get the value of the derivative at x = 0:

$f'(0)=\frac{e^0(11-15(0))}{(5(0)-2)^2}$ which simplifies to

$f'(0)=\frac{11}{4}$ which translates to

The slope of the function is 11/4 at the point (0, -1/2)