dg/dt = ?
Understand the Problem
The question is asking for the derivative of the given function g with respect to time t. It involves differentiating the sine function and applying the chain rule due to the function's composition.
Answer
The derivative is: $$ \frac{dg}{dt} = \cos\left(\frac{t}{\sqrt{t + 3}}\right) \cdot \frac{t + 6}{2(t + 3)\sqrt{t + 3}} $$
Answer for screen readers
The derivative of ( g ) with respect to ( t ) is:
$$ \frac{dg}{dt} = \cos\left(\frac{t}{\sqrt{t + 3}}\right) \cdot \frac{t + 6}{2(t + 3)\sqrt{t + 3}} $$
Steps to Solve
-
Identify the inner and outer functions
In the function ( g = \sin\left(\frac{t}{\sqrt{t + 3}}\right) ), the outer function is ( \sin(u) ) where ( u = \frac{t}{\sqrt{t + 3}} ).
-
Differentiate the outer function
The derivative of the sine function is:
$$ \frac{dg}{du} = \cos(u) $$
-
Differentiate the inner function
We need to differentiate ( u = \frac{t}{\sqrt{t + 3}} ). Use the quotient rule, which states that if ( u = \frac{f(t)}{g(t)} ), then:
$$ u' = \frac{f'g - fg'}{g^2} $$
Here, ( f(t) = t ) and ( g(t) = \sqrt{t + 3} ). Thus, ( f' = 1 ) and ( g' = \frac{1}{2\sqrt{t + 3}} ).
Substituting into the quotient rule:
$$ u' = \frac{1 \cdot \sqrt{t + 3} - t \cdot \frac{1}{2\sqrt{t + 3}}}{(t + 3)} $$
-
Simplify the derivative of the inner function
Simplify the expression:
$$ u' = \frac{\sqrt{t + 3} - \frac{t}{2\sqrt{t + 3}}}{t + 3} = \frac{2(t + 3) - t}{2(t + 3)\sqrt{t + 3}} = \frac{t + 6}{2(t + 3)\sqrt{t + 3}} $$
-
Apply the chain rule
Now apply the chain rule:
$$ \frac{dg}{dt} = \frac{dg}{du} \cdot \frac{du}{dt} = \cos\left(\frac{t}{\sqrt{t + 3}}\right) \cdot \frac{t + 6}{2(t + 3)\sqrt{t + 3}} $$
The derivative of ( g ) with respect to ( t ) is:
$$ \frac{dg}{dt} = \cos\left(\frac{t}{\sqrt{t + 3}}\right) \cdot \frac{t + 6}{2(t + 3)\sqrt{t + 3}} $$
More Information
This derivative represents how the function ( g ) changes with time ( t ). The use of the chain rule is essential since the function's structure involves a composition of functions (sine and a fractional function).
Tips
- Forgetting to apply the chain rule can lead to an incorrect derivative.
- Misapplying the quotient rule or simplifying incorrectly can yield an erroneous result. Always double-check each simplification step.
AI-generated content may contain errors. Please verify critical information
