Find dx/dt x=t(2t+1)^2

Problem

d()/d(t)*t*(2*t+1)^2

Solution

1. Identify the rule needed for differentiation. Since the expression is a product of t and (2*t+1)^2 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=(2*t+1)^2

3. Differentiate u with respect to t

d(u)/d(t)=1

4. Differentiate v with respect to t using the chain rule.

d(v)/d(t)=2*(2*t+1)⋅2

d(v)/d(t)=4*(2*t+1)

5. Apply the product rule formula.

d(x)/d(t)=t*(4*(2*t+1))+(2*t+1)^2*(1)

6. Factor out the common term (2*t+1) to simplify the expression.

d(x)/d(t)=(2*t+1)*(4*t+(2*t+1))

7. Combine like terms inside the parentheses.

d(x)/d(t)=(2*t+1)*(6*t+1)

Final Answer

d(x)/d(t)=(2*t+1)*(6*t+1)

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