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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 2y2(2x2y2x2+y32y2+2)

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

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

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

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

y(2x2y)+y(2)y(2x2)y(2y2)+y(y3)=2x2y(y)+y(2)y(2x2)y(2y2)+y(y3)

Apply the power rule y(yn)=ny1+n with n=1, in other words y(y)=1:

2x2y(y)+y(2)y(2x2)y(2y2)+y(y3)=2x21+y(2)y(2x2)y(2y2)+y(y3)

The derivative of a constant is 0:

2x2y(2x2)+y(2)y(2y2)+y(y3)=2x2(0)+y(2)y(2y2)+y(y3)

Apply the constant multiple rule y(cf)=cy(f) with c=2 and f=y2:

2x2y(2y2)+y(2)+y(y3)=2x2(2y(y2))+y(2)+y(y3)

Apply the power rule y(yn)=ny1+n with n=2:

2x22y(y2)+y(2)+y(y3)=2x22(2y1+2)+y(2)+y(y3)=2x24y+y(2)+y(y3)

The derivative of a constant is 0:

2x24y+y(2)+y(y3)=2x24y+(0)+y(y3)

Apply the power rule y(yn)=ny1+n with n=3:

2x24y+y(y3)=2x24y+(3y1+3)=2x2+3y24y

Thus, y(2x2y2x2+y32y2+2)=2x2+3y24y

Next, 2y2(2x2y2x2+y32y2+2)=y(y(2x2y2x2+y32y2+2))=y(2x2+3y24y)

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

y(2x2+3y24y)=(y(2x2)y(4y)+y(3y2))

Apply the constant multiple rule y(cf)=cy(f) with c=3 and f=y2:

y(3y2)+y(2x2)y(4y)=(3y(y2))+y(2x2)y(4y)

Apply the power rule y(yn)=ny1+n with n=2:

3y(y2)+y(2x2)y(4y)=3(2y1+2)+y(2x2)y(4y)=6y+y(2x2)y(4y)

Apply the constant multiple rule y(cf)=cy(f) with c=4 and f=y:

6yy(4y)+y(2x2)=6y(4y(y))+y(2x2)

Apply the power rule y(yn)=ny1+n with n=1, in other words y(y)=1:

6y4y(y)+y(2x2)=6y41+y(2x2)

The derivative of a constant is 0:

6y4+y(2x2)=6y4+(0)

Thus, y(2x2+3y24y)=6y4

Therefore, 2y2(2x2y2x2+y32y2+2)=6y4

Answer: 2y2(2x2y2x2+y32y2+2)=6y4