1. The problem asks to find the partial derivative of the function $f(x,y,t) = x^2 y^3 t^4 - x y^4 t^2 + x^4 t^5$ with respect to $t$, and then evaluate it at the point $(1,1,1)$. 2. The formula for the partial derivative with respect to $t$ is to differentiate $f$ treating $x$ and $y$ as constants: $$f_t = \frac{\partial}{\partial t} \left(x^2 y^3 t^4 - x y^4 t^2 + x^4 t^5\right)$$ 3. Differentiate each term: - For $x^2 y^3 t^4$, derivative w.r.t. $t$ is $4 x^2 y^3 t^3$ - For $- x y^4 t^2$, derivative w.r.t. $t$ is $-2 x y^4 t$ - For $x^4 t^5$, derivative w.r.t. $t$ is $5 x^4 t^4$ 4. So, $$f_t = 4 x^2 y^3 t^3 - 2 x y^4 t + 5 x^4 t^4$$ 5. Now evaluate at $(x,y,t) = (1,1,1)$: $$f_t(1,1,1) = 4 \cdot 1^2 \cdot 1^3 \cdot 1^3 - 2 \cdot 1 \cdot 1^4 \cdot 1 + 5 \cdot 1^4 \cdot 1^4 = 4 - 2 + 5 = 7$$ 6. Therefore, the value of $f_t(1,1,1)$ is 7.