find the derivative of the following function.\n\n$f(t)=t^{\\frac{3}{2}}e^{t}$\n\n$f(t)=\\square$
find the derivative of the following function.\n\n$f(t)=t^{\\frac{3}{2}}e^{t}$\n\n$f(t)=\\square$
Answer
Explanation:
Step1: Apply the product rule
The product rule states that if (y = u\cdot v), then (y^\prime=u^\prime v + uv^\prime). Let (u = t^{\frac{3}{2}}) and (v = e^{t}). First, find (u^\prime): Using the power rule ((x^{n})^\prime=nx^{n - 1}), we have (u^\prime=\frac{3}{2}t^{\frac{3}{2}-1}=\frac{3}{2}t^{\frac{1}{2}}). And (v^\prime=(e^{t})^\prime = e^{t}).
Step2: Substitute into the product rule formula
(f^\prime(t)=u^\prime v+uv^\prime=\frac{3}{2}t^{\frac{1}{2}}e^{t}+t^{\frac{3}{2}}e^{t}). Factor out (t^{\frac{1}{2}}e^{t}): (f^\prime(t)=t^{\frac{1}{2}}e^{t}(\frac{3}{2}+t)).
Answer:
(t^{\frac{1}{2}}e^{t}(\frac{3}{2}+t))