Find the Derivative - d/dt t square root of 16-t^2

Problem

d()/d(t)*t√(,16−t2)

Solution

1. Identify the rule needed for the expression, which is a product of t and √(,16−t2) We must use the product rule: (d(u)*v)/d(t)=ud(v)/d(t)+vd(u)/d(t)

2. Assign the variables for the product rule: let u=t and v=(16−t2)(1/2)

3. Differentiate u with respect to t

d(u)/d(t)=1

4. Differentiate v using the chain rule:

d(v)/d(t)=1/2*(16−t2)(−1/2)⋅(−2*t)

5. Simplify the derivative of v

d(v)/d(t)=(−t)/√(,16−t2)

6. Apply the product rule formula:

d()/d(t)*t√(,16−t2)=t⋅(−t)/√(,16−t2)+√(,16−t2)⋅1

7. Combine the terms into a single expression:

d()/d(t)*t√(,16−t2)=(−t2)/√(,16−t2)+√(,16−t2)

8. Find a common denominator to simplify the result:

d()/d(t)*t√(,16−t2)=(−t2+(16−t2))/√(,16−t2)

9. Simplify the numerator:

d()/d(t)*t√(,16−t2)=(16−2*t2)/√(,16−t2)

Final Answer

d()/d(t)*t√(,16−t2)=(16−2*t2)/√(,16−t2)

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