Find the Derivative - d/dt t square root of 4-t

Problem

d()/d(t)*t√(,4−t)

Solution

1. Identify the rule needed for the expression, which is the product of two functions: u(t)=t and v(t)=√(,4−t)

2. Apply the product rule, which states (d(u)*v)/d(t)=ud(v)/d(t)+vd(u)/d(t)

3. Differentiate the first part u(t)=t to get d(t)/d(t)=1

4. Differentiate the second part v(t)=√(,4−t) using the chain rule.

d(√(,4−t))/d(t)=1/(2√(,4−t))⋅(−1)

d(√(,4−t))/d(t)=−1/(2√(,4−t))

5. Combine the parts using the product rule formula.

d()/d(t)*t√(,4−t)=t*(−1/(2√(,4−t)))+√(,4−t)*(1)

6. Simplify the expression by finding a common denominator.

d()/d(t)*t√(,4−t)=−t/(2√(,4−t))+(2*(4−t))/(2√(,4−t))

d()/d(t)*t√(,4−t)=(−t+8−2*t)/(2√(,4−t))

d()/d(t)*t√(,4−t)=(8−3*t)/(2√(,4−t))

Final Answer

d()/d(t)*t√(,4−t)=(8−3*t)/(2√(,4−t))

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