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The multivariable chain rule practice problems

dz/dt and ∂z/∂s when the variables themselves depend on other variables. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.

Let \( \displaystyle z = x^{2} - 3 x y - 3 y^{2} \) with \( \displaystyle x = s^{2} - t^{2} \), \( \displaystyle y = 2 s t \). Find \( \displaystyle \frac{\partial z}{\partial s} \) at \( \displaystyle s = 2 \), \( \displaystyle t = 0 \).

Problem 10.339easy✓ Every equation proved

Let \( \displaystyle z = x^{2} - 2 x y - 3 y^{2} \) with \( \displaystyle x = 2 t + 1 \), \( \displaystyle y = t^{2} - 1 \). Use the chain rule to find \( \displaystyle \frac{dz}{dt} \) at \( \displaystyle t = 1 \).

Problem 10.340easy✓ Every equation proved

Let \( \displaystyle z = x^{2} y + 3 y^{3} \) with \( \displaystyle x = \cos{\left(t \right)} \), \( \displaystyle y = \sin{\left(t \right)} \). Use the chain rule to find \( \displaystyle \frac{dz}{dt} \) at \( \displaystyle t = 0 \).

Problem 10.341easy✓ Every equation proved

Let \( \displaystyle z = x e^{- 2 y} \) with \( \displaystyle x = s^{2} - t^{2} \), \( \displaystyle y = 2 s t \). Find \( \displaystyle \frac{\partial z}{\partial s} \) at \( \displaystyle s = 1 \), \( \displaystyle t = 1 \).

Problem 10.342easy✓ Every equation proved

Let \( \displaystyle z = x^{2} + 3 x y - 3 y^{2} \) with \( \displaystyle x = t^{2} \), \( \displaystyle y = t^{3} \). Use the chain rule to find \( \displaystyle \frac{dz}{dt} \) at \( \displaystyle t = 1 \).

Problem 10.343easy✓ Every equation proved

Let \( \displaystyle z = x^{2} y - 2 y^{3} \) with \( \displaystyle x = s^{2} - t^{2} \), \( \displaystyle y = 2 s t \). Find \( \displaystyle \frac{\partial z}{\partial s} \) at \( \displaystyle s = 1 \), \( \displaystyle t = 0 \).

Problem 10.344easy✓ Every equation proved

Let \( \displaystyle z = x^{2} y - y^{3} \) with \( \displaystyle x = t^{2} \), \( \displaystyle y = t^{3} \). Use the chain rule to find \( \displaystyle \frac{dz}{dt} \) at \( \displaystyle t = 1 \).

Problem 10.346easy✓ Every equation proved

Let \( \displaystyle z = \ln{\left(x^{2} + y^{2} + 1 \right)} \) with \( \displaystyle x = s t \), \( \displaystyle y = s + t \). Find \( \displaystyle \frac{\partial z}{\partial s} \) at \( \displaystyle s = 2 \), \( \displaystyle t = 0 \).

Problem 10.347easy✓ Nihil obstat

Let \( \displaystyle z = x^{2} y - 3 y^{3} \) with \( \displaystyle x = s \cos{\left(t \right)} \), \( \displaystyle y = s \sin{\left(t \right)} \). Find \( \displaystyle \frac{\partial z}{\partial s} \) at \( \displaystyle s = 1 \), \( \displaystyle t = 0 \).

Problem 10.348easy✓ Every equation proved

Let \( \displaystyle z = x \cos{\left(y \right)} + y e^{x} \) with \( \displaystyle x = s t \), \( \displaystyle y = s + t \). Find \( \displaystyle \frac{\partial z}{\partial s} \) at \( \displaystyle s = 1 \), \( \displaystyle t = 1 \).

Problem 10.345medium✓ Nihil obstat