Find the Derivative - d/d@VAR f(t)=10te^(-5/2t)

Combine ${\displaystyle t}$ and ${\displaystyle \frac{5}{2}}$.

Move ${\displaystyle 5}$ to the left of ${\displaystyle t}$.

Since ${\displaystyle 10}$ is constant with respect to ${\displaystyle t}$, the derivative of ${\displaystyle 10t{e}^{-\frac{5t}{2}}}$ with respect to ${\displaystyle t}$ is ${\displaystyle 10\frac{d}{dt}\left[t{e}^{-\frac{5t}{2}}\right]}$.

To apply the Chain Rule, set ${\displaystyle u}$ as ${\displaystyle -\frac{5t}{2}}$.

Differentiate using the Exponential Rule which states that ${\displaystyle \frac{d}{du}\left[a^u\right]}$ is ${\displaystyle a^u\ln \left(a\right)}$ where ${\displaystyle a}$=${\displaystyle e}$.

Replace all occurrences of ${\displaystyle u}$ with ${\displaystyle -\frac{5t}{2}}$.

Since ${\displaystyle -\frac{5}{2}}$ is constant with respect to ${\displaystyle t}$, the derivative of ${\displaystyle -\frac{5t}{2}}$ with respect to ${\displaystyle t}$ is ${\displaystyle -\frac{5}{2}\frac{d}{dt}\left[t\right]}$.

Combine fractions.

Differentiate using the Power Rule which states that ${\displaystyle \frac{d}{dt}\left[t^n\right]}$ is ${\displaystyle n{t}^{n-1}}$ where ${\displaystyle n=1}$.

Multiply ${\displaystyle -1}$ by ${\displaystyle 1}$.

Differentiate using the Power Rule which states that ${\displaystyle \frac{d}{dt}\left[t^n\right]}$ is ${\displaystyle n{t}^{n-1}}$ where ${\displaystyle n=1}$.

Multiply ${\displaystyle e^{-\frac{5t}{2}}}$ by ${\displaystyle 1}$.

Apply the distributive property.

Combine terms.

Reorder terms.

Reorder factors in ${\displaystyle -25e^{-\frac{5t}{2}}t+10e^{-\frac{5t}{2}}}$.