Determine the gradient of the function f(x,y)=5x-x^5*y^6 at the point (-1,1)

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll have to determine the gradient vector at the point (-1 , 1).

Since the function is of 2 variables, the gradient of the function is the vector function that is calculated using the formula below:

Grad f(x,y) = [df(x,y)/dx]*i + [df(x,y)/dy]*j

[df(x,y)/dx] and [df(x,y)/dy] are the partial derivatives of the function.

We'll calculate the partial derivative of f(x,y), with respect to x, assuming that y is constant.

df(x,y)/dx = 5 - 5x^4*y^6 = 5(1 - x^4*y^6)

We'll calculate the partial derivative of f(x,y), with respect to y, assuming that x is constant.

df(x,y)/dx = - 6y^5*x^5

Grad f(x,y) = 5(1 - x^4*y^6)*i - 6y^5*x^5*j

Grad f(-1,1) = 5(1 - 1*1)*i - 6*1^5*(-1)^5*j

Grad f(-1,1) = 0*i + 6*j

The gradient of the function f(x,y)=5x-x^5*y^6, at the point (-1,1), is Grad f(-1,1) = 6*j.

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