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Calculatrice de dérivées partielles

Calculer les dérivées partielles étape par étape

Cette calculatrice en ligne calculera la dérivée partielle de la fonction, avec les étapes indiquées. Vous pouvez spécifier n'importe quel ordre d'intégration.

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Hint: type x^2,y to calculate , or enter x,y^2,x to find .

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Solution

Your input: find 2x2(2x2y2x2+y32y2+2)

First, find x(2x2y2x2+y32y2+2)

The derivative of a sum/difference is the sum/difference of derivatives:

x(2x2y2x2+y32y2+2)=(x(2)x(2x2)x(2y2)+x(y3)+x(2x2y))

Apply the constant multiple rule x(cf)=cx(f) with c=2y and f=x2:

x(2x2y)+x(2)x(2x2)x(2y2)+x(y3)=2yx(x2)+x(2)x(2x2)x(2y2)+x(y3)

Apply the power rule x(xn)=nx1+n with n=2:

2yx(x2)+x(2)x(2x2)x(2y2)+x(y3)=2y(2x1+2)+x(2)x(2x2)x(2y2)+x(y3)=4xy+x(2)x(2x2)x(2y2)+x(y3)

Apply the constant multiple rule x(cf)=cx(f) with c=2 and f=x2:

4xyx(2x2)+x(2)x(2y2)+x(y3)=4xy(2x(x2))+x(2)x(2y2)+x(y3)

Apply the power rule x(xn)=nx1+n with n=2:

4xy2x(x2)+x(2)x(2y2)+x(y3)=4xy2(2x1+2)+x(2)x(2y2)+x(y3)=4xy4x+x(2)x(2y2)+x(y3)

The derivative of a constant is 0:

4xy4xx(2y2)+x(2)+x(y3)=4xy4x(0)+x(2)+x(y3)

The derivative of a constant is 0:

4xy4x+x(2)+x(y3)=4xy4x+(0)+x(y3)

The derivative of a constant is 0:

4xy4x+x(y3)=4xy4x+(0)=4x(y1)

Thus, x(2x2y2x2+y32y2+2)=4x(y1)

Next, 2x2(2x2y2x2+y32y2+2)=x(x(2x2y2x2+y32y2+2))=x(4x(y1))

Apply the constant multiple rule x(cf)=cx(f) with c=4(y1) and f=x:

x(4x(y1))=4(y1)x(x)

Apply the power rule x(xn)=nx1+n with n=1, in other words x(x)=1:

4(y1)x(x)=4(y1)1=4y4

Thus, x(4x(y1))=4y4

Therefore, 2x2(2x2y2x2+y32y2+2)=4y4

Answer: 2x2(2x2y2x2+y32y2+2)=4y4