Questions: Find the derivative of the following function. y=(2t-1)(5t-5)^-1 dy/dt=

Transcript text: Find the derivative of the following function. \[ \begin{array}{l} y=(2 t-1)(5 t-5)^{-1} \\ \frac{d y}{d t}=\square \end{array} \]

Solution

Solution Steps

Step 1: Define the function and its derivative

Given the function $y = (2 t - 1)(5 t - 5)^{-1}$, we use the quotient rule for differentiation: $\frac{dy}{dt} = \frac{u'(t) v(t) - u(t) v'(t)}{[v(t)]^2}$, where $u(t) = (a t - b)$ and $v(t) = (c t - d)$.

Step 2: Calculate the derivatives of $u(t)$ and $v(t)$

Given $u(t) = (2 t - 1)$ and $v(t) = (5 t - 5)$, the derivatives are $u'(t) = 2$ and $v'(t) = 5$ respectively.

Step 3: Apply the quotient rule

Substituting into the quotient rule, we get $\frac{dy}{dt} = \frac{2_(5_t - 5) - (2_t - 1)_5}{(5*t - 5)^2}$.

Step 4: Simplify the expression

Simplifying, we get $\frac{dy}{dt} = \frac{10_t - 10 - 10_t + 5}{(5*t - 5)^2}$.

Step 5: Further simplification

Further simplifying, we get $\frac{dy}{dt} = \frac{-5}{(5*t - 5)^2}$.

Final Answer:

The derivative of the function with respect to t is $\frac{dy}{dt} = -0.2$, or in simplified form, $\frac{dy}{dt} = \frac{-5}{(5*t - 5)^2}$ rounded to 2 decimal places.

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