Questions: If (f(t)=(t^2+5 t+4)(7 t^2+4)), find (f^prime(t)). ((28 t^3+105 t^2+64 t+20)) Find (f^prime(3))

Transcript text: If $f(t)=\left(t^{2}+5 t+4\right)\left(7 t^{2}+4\right)$, find $f^{\prime}(t)$. \[ \left(28 t^{3}+105 t^{2}+64 t+20\right) \] Find $f^{\prime}(3)$

Solution

Solution Steps

To find the derivative $ f'(t) $ of the function $ f(t) = (t^2 + 5t + 4)(7t^2 + 4) $, we will use the product rule of differentiation. The product rule states that if you have a function $ f(t) = u(t) \cdot v(t) $, then the derivative $ f'(t) = u'(t) \cdot v(t) + u(t) \cdot v'(t) $. Here, $ u(t) = t^2 + 5t + 4 $ and $ v(t) = 7t^2 + 4 $. We will first find the derivatives $ u'(t) $ and $ v'(t) $, and then apply the product rule. After finding $ f'(t) $, we will evaluate it at $ t = 3 $ to find $ f'(3) $.

Step 1: Define the Functions and Their Derivatives

Given the function $ f(t) = (t^2 + 5t + 4)(7t^2 + 4) $, we define:

$ u(t) = t^2 + 5t + 4 $
$ v(t) = 7t^2 + 4 $

The derivatives are:

$ u'(t) = \frac{d}{dt}(t^2 + 5t + 4) = 2t + 5 $
$ v'(t) = \frac{d}{dt}(7t^2 + 4) = 14t $

Step 2: Apply the Product Rule

Using the product rule for differentiation, $ f'(t) = u'(t) \cdot v(t) + u(t) \cdot v'(t) $, we have: \[ f'(t) = (2t + 5)(7t^2 + 4) + (t^2 + 5t + 4)(14t) \]

Step 3: Simplify the Expression

Expanding and simplifying the expression for $ f'(t) $: \[ f'(t) = 14t(t^2 + 5t + 4) + (2t + 5)(7t^2 + 4) \]

Step 4: Evaluate $ f'(t) $ at $ t = 3 $

Substitute $ t = 3 $ into the expression for $ f'(t) $: \[ f'(3) = 14 \cdot 3(3^2 + 5 \cdot 3 + 4) + (2 \cdot 3 + 5)(7 \cdot 3^2 + 4) \] \[ f'(3) = 14 \cdot 3(9 + 15 + 4) + (6 + 5)(63 + 4) \] \[ f'(3) = 14 \cdot 3 \cdot 28 + 11 \cdot 67 \] \[ f'(3) = 1176 + 737 = 1913 \]