Mathematics

Problem

A differentiable function of time is given by

F(t)=e^{5t}\,\bigl(1+a t^{2}\bigr)^{b},

where

a

and

b

are constants with

a>0

and

b\neq0

. (a) Compute

\dfrac{dF}{dt}

in terms of

t

a

, and

b

, leaving your answer factored. (b) Evaluate

\dfrac{dF}{dt}

t=0

and interpret the result in terms of how

F

initially changes with time. (c) Holding

t

fixed, determine

\dfrac{\partial F}{\partial a}

and describe how increasing

a

affects

F(t)

for a fixed instant

t>0

. Justify your reasoning.

Intuitive explanation

The key is recognizing that you have a product of two functions, so the product rule is your main tool. The exponential $e^{5t}$ has a simple derivative, while $(1+at^2)^b$ requires the chain rule. Once you differentiate, you'll likely factor out the common exponential term to simplify and reveal the structure of $F'(t)$ .

Intuitive explanation

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