Obtain ∂𝑣∂𝑥 if 𝑉=𝑓(𝑥2+𝑦2)Question 3Answera.𝑓(2𝑥+𝑦2)b.2𝑥𝑓'(𝑥2+𝑦2)c.2𝑓'(𝑥+𝑦2)d.2𝑥𝑓(𝑥2+2𝑦2)

Question

Obtain ∂𝑣∂𝑥 if 𝑉=𝑓(𝑥2+𝑦2)Question 3Answera.𝑓(2𝑥+𝑦2)b.2𝑥𝑓'(𝑥2+𝑦2)c.2𝑓'(𝑥+𝑦2)d.2𝑥𝑓(𝑥2+2𝑦2)

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Solution

Para obtener ∂𝑣/∂𝑥 si 𝑉=𝑓(𝑥²+𝑦²), seguimos estos pasos:

Definimos 𝑢 = 𝑥² + 𝑦².
Entonces, 𝑉 = 𝑓(𝑢).

Para encontrar la derivada parcial de 𝑉 con respecto a 𝑥, usamos la regla de la cadena:

∂𝑉/∂𝑥 = ∂𝑉/∂𝑢 * ∂𝑢/∂𝑥.

Primero, encontramos ∂𝑢/∂𝑥:

∂𝑢/∂𝑥 = ∂(𝑥² + 𝑦²)/∂𝑥 = 2𝑥.

Luego, encontramos ∂𝑉/∂𝑢:

∂𝑉/∂𝑢 = 𝑓'(𝑢) = 𝑓'(𝑥² + 𝑦²).

Finalmente, combinamos estos resultados:

∂𝑉/∂𝑥 = 𝑓'(𝑥² + 𝑦²) * 2𝑥.

Por lo tanto, la respuesta correcta es:

b. 2𝑥𝑓'(𝑥²+𝑦²).

This problem has been solved

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