Question

Find derivative of

a.) f²(x)={-21 e⁷x}+14 e²x / (e⁵x}+1²
b.) f²(x)={49 e⁷x}+14 e²x / (e⁵x}+1

Answer 1

a.) The derivative of f²(x) is
`\[ f'(x) = (-147e^(7x) + 28e^(2x))/((e^(5x) + 1)^2) \]`

b.) The derivative of f²(x) is
`\[ f'(x) = (343e^(7x) + 28e^(2x))/((e^(5x) + 1)^2) \]`

Step-by-step explanation:

In both cases, we use the quotient rule to find the derivative. The quotient rule states that for a function
`\[ g(x) = (u(x))/(v(x)) \]`, the derivative
`\[ g'(x) \]` is given by
`\[ g'(x) = (u'v - uv')/(v^2) \]`.

For option a:

`\[ u(x) = -21e^(7x) + 14e^(2x) \]`

`\[ v(x) = e^(5x) + 1 \]`

`\[ u'(x) = -147e^(7x) + 28e^(2x) \]`

`\[ v'(x) = 5e^(5x) \]`

Using the quotient rule, we get the derivative
`\[ f'(x) = (u'v - uv')/(v^2) = (-147e^(7x) + 28e^(2x))/((e^(5x) + 1)^2) \].`

For option b:

`\[ u(x) = 49e^(7x) + 14e^(2x) \]`

`\[ v(x) = e^(5x) + 1 \]`

`\[ u'(x) = 343e^(7x) + 28e^(2x) \]`

\
`[ v'(x) = 5e^(5x) \]`

Using the quotient rule, we get the derivative
`\[ f'(x) = (u'v - uv')/(v^2) = (343e^(7x) + 28e^(2x))/((e^(5x) + 1)^2) \].`

These derivatives provide the rate at which the given functions change with respect to the variable x.