Question

Find all the second partial derivatives of t = e⁽⁻⁶ʳ⁾ cos(θ)?

Answer 1

To find all second partial derivatives of the function t = e^(-6r) cos(θ), we differentiate twice with respect to r and θ respectively, obtaining four second partial derivatives including the mixed partials.

Step-by-step explanation:

To find all the second partial derivatives of the function t = e−6r cos(θ), we need to differentiate it twice with respect to r and θ respectively. Here are the steps for calculating each:

• First differentiate t with respect to r, which gives us the first partial derivative with respect to r: (∂t/∂r) = -6e−6r cos(θ).
• Now, differentiate (∂t/∂r) with respect to r again to find the second partial derivative with respect to r: (∂2t/∂r2) = 36e−6r cos(θ).
• Next, differentiate t with respect to θ, giving the first partial derivative with respect to θ: (∂t/∂θ) = -e−6r sin(θ).
• Finally, differentiate (∂t/∂θ) with respect to θ for the second partial derivative with respect to θ: (∂2t/∂θ2) = -e−6r cos(θ).

Additionally, we have to consider the mixed partial derivatives. Differentiate (∂t/∂r) with respect to θ and vice versa to find that both are equal to 6e−6r sin(θ).