Question 1: What Is FT T4LN3T 5?

Question 1 If Ft T4ln3t 5 What Is F T 4t33 3t 5

Question 1 · If f(t) = t4ln(3t + 5), what is f '(t)?

· · · 4t3[3 / (3t + 5)]

· · · 4t3 / (3t + 5)

· · · (t4)[ 3 / (3t + 5) ] + [ln(3t + 5)](4t3)

· · · (t4)(4t3) + [ln(3t + 5)][ 3 / (3t + 5) ]

· · · None of These

Paper For Above instruction

The problem involves differentiating the function f(t) = t^4 ln(3t + 5). To find the derivative f'(t), we apply the product rule, as the function is a product of t^4 and ln(3t + 5). The product rule states that if f(t) = u(t) v(t), then f'(t) = u'(t) v(t) + u(t) * v'(t).

First, define u(t) = t^4 and v(t) = ln(3t + 5). The derivative of u(t) is straightforward: u'(t) = 4t^3. The derivative of v(t) requires the chain rule: v'(t) = (1 / (3t + 5)) * 3 = 3 / (3t + 5).

Applying the product rule, we have:

f'(t) = u'(t) v(t) + u(t) v'(t) = 4t^3 ln(3t + 5) + t^4 (3 / (3t + 5)).

Therefore, the correct expression for the derivative is:

f'(t) = 4t^3 ln(3t + 5) + (t^4 * 3) / (3t + 5).

This corresponds to the choice: (t^4) [3 / (3t + 5)] + [ln(3t + 5)](4t^3), which matches the product rule expansion.

In conclusion, the derivative involves applying both the product rule and the chain rule appropriately, resulting in the combined expression above.

References

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