Find the Derivative - d/dt t square root of 9-t^2

Problem

d()/d(t)*t√(,9−t2)

Solution

1. Identify the rule needed for the derivative. Since the expression is a product of t and √(,9−t2) we must use the product rule: (d(u)*v)/d(t)=ud(v)/d(t)+vd(u)/d(t)

2. Assign the parts of the product. Let u=t and v=√(,9−t2)=(9−t2)(1/2)

3. Differentiate u with respect to t

d(t)/d(t)=1

4. Differentiate v using the chain rule.

d(9−t2)/d(t)=1/2*(9−t2)(−1/2)⋅(−2*t)

5. Simplify the derivative of v

d(v)/d(t)=(−t)/√(,9−t2)

6. Apply the product rule formula.

(d(t)√(,9−t2))/d(t)=t⋅(−t)/√(,9−t2)+√(,9−t2)⋅1

7. Combine the terms into a single fraction by finding a common denominator.

(d(t)√(,9−t2))/d(t)=(−t2)/√(,9−t2)+(9−t2)/√(,9−t2)

8. Simplify the numerator.

(d(t)√(,9−t2))/d(t)=(9−2*t2)/√(,9−t2)

Final Answer

(d(t)√(,9−t2))/d(t)=(9−2*t2)/√(,9−t2)

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