Question

Calculus early transcendental functions. Find the derivative using the quotient rule

Answer 1

The quotient rule is a formula used to find the derivative of a function that is the quotient of two other functions.

Step-by-step explanation:

The quotient rule is a formula used to find the derivative of a function that is the quotient of two other functions. The formula is as follows:

When differentiating a function f(x) divided by g(x), the derivative is given by:

[(derivative of f(x)) * (g(x)) - (f(x)) * (derivative of g(x))] / [(g(x))^2]

To use the quotient rule, you need to find the derivatives of both the numerator and the denominator, substitute these values into the formula, and simplify.

Answer 2

`f^(\prime)(t)=(-14)/(t^3)`

Step-by-step explanation:

`\begin{gathered} f(t)\text{ = }(7)/(t^2)\text{ + }(4)/(t^5) \\ finding\text{ the LCM:} \\ f(t)\text{ = }\frac{7t^3+\text{ 4}}{t^5} \end{gathered}`

Using quotient rule:

`\begin{gathered} f^(\prime)(t)\text{ = }\frac{\text{v}\frac{du}{d\text{ t}}\text{ - u }\frac{dv}{d\text{ t}}}{v^2} \\ u=7t^3+4,du/dt=3(7)t^2=21t^2 \\ v=t^5,dv/dt=5t^4 \\ \\ f^(\prime)(t)\text{ = }(d)/(dt)(\frac{7t^3+\text{ 4}}{t^5}) \end{gathered}`
`\begin{gathered} f^(\prime)(t)\text{ = }(t^5(21t^2)-7t^3(5t^4))/((t^5)^2) \\ f^(\prime)(t)\text{ = }\frac{21t^7-35t^7}{t^(10)^{}} \end{gathered}`
`\begin{gathered} f^(\prime)(t)\text{ = }(t^7(21-35))/(t^(10)) \\ f^(\prime)(t)\text{ = }t^7* t^(-10)*(21-35) \\ f^(\prime)(t)=t^(-3)(21-35)\text{ }=t^(-3)(-14) \\ f^(\prime)(t)=(-14)/(t^3) \end{gathered}`