Find the derivative of y=t^2(5t-8)^4.
To find the derivative of the function y = t^2(5t - 8)^4, we can use the product and chain rules of differentiation.
Step 1: Apply the product rule.
The product rule states that if we have two functions, u(t) and v(t), their derivative can be found using the formula:
(d/dt)(u(t) * v(t)) = u'(t) * v(t) + u(t) * v'(t).
In this case, let's define u(t) = t^2 and v(t) = (5t - 8)^4.
Then, u'(t) = 2t and v'(t) = 4(5t - 8)^3.
Applying the product rule, we have:
(d/dt)(t^2(5t - 8)^4) = u'(t) * v(t) + u(t) * v'(t)
= 2t * (5t - 8)^4 + t^2 * 4(5t - 8)^3.
Step 2: Simplify the expression.
To simplify the expression, we can expand the powers of (5t - 8) and distribute the terms.
(d/dt)(t^2(5t - 8)^4) = 2t * (5t - 8)^4 + t^2 * 4(5t - 8)^3
= 2t * 625t^4 - 800t^3 + 320t^2 - 256t + t^2 * 4 * 125t^3 - 4 * 200t^2 + 4 * 64t
= 1250t^5 - 1600t^4 + 640t^3 - 512t + 500t^5 - 1000t^4 + 256t - 800t^3 + 320t^2 - 256t.
Step 3: Simplify further.
Combine like terms to simplify the expression:
(d/dt)(t^2(5t - 8)^4) = 1750t^5 - 2600t^4 + 760t^3 - 512t + 320t^2.
Therefore, the derivative of y = t^2(5t - 8)^4 is dy/dt = 1750t^5 - 2600t^4 + 760t^3 - 512t + 320t^2.