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  2. Differenziazione Implicita

Calcolatrice di Differenziazione implicita

Risolvete i vostri problemi di matematica con la nostra calcolatrice Differenziazione implicita passo-passo. Migliorate le vostre abilitΓ  matematiche con il nostro ampio elenco di problemi impegnativi. Trova tutte le nostre calcolatrici qui.

ddx (x2+y2=16)
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1

Here, we show you a step-by-step solved example of implicit differentiation. This solution was automatically generated by our smart calculator:

ddx(x2+y2=16)\frac{d}{dx}\left(x^2+y^2=16\right)
2

Apply implicit differentiation by taking the derivative of both sides of the equation with respect to the differentiation variable

ddx(x2+y2)=ddx(16)\frac{d}{dx}\left(x^2+y^2\right)=\frac{d}{dx}\left(16\right)
3

The derivative of the constant function (1616) is equal to zero

ddx(x2+y2)=0\frac{d}{dx}\left(x^2+y^2\right)=0
4

The derivative of a sum of two or more functions is the sum of the derivatives of each function

ddx(x2)+ddx(y2)=0\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(y^2\right)=0

The power rule for differentiation states that if nn is a real number and f(x)=xnf(x) = x^n, then fβ€²(x)=nxnβˆ’1f'(x) = nx^{n-1}

ddx(x2)+2y2βˆ’1ddx(y)=0\frac{d}{dx}\left(x^2\right)+2y^{2-1}\frac{d}{dx}\left(y\right)=0

Add the values 22 and βˆ’1-1

ddx(x2)+2y1ddx(y)=0\frac{d}{dx}\left(x^2\right)+2y^{1}\frac{d}{dx}\left(y\right)=0

The power rule for differentiation states that if nn is a real number and f(x)=xnf(x) = x^n, then fβ€²(x)=nxnβˆ’1f'(x) = nx^{n-1}

2y2βˆ’1ddx(y)2y^{2-1}\frac{d}{dx}\left(y\right)

Subtract the values 22 and βˆ’1-1

2y1ddx(y)2y^{1}\frac{d}{dx}\left(y\right)
5

The power rule for differentiation states that if nn is a real number and f(x)=xnf(x) = x^n, then fβ€²(x)=nxnβˆ’1f'(x) = nx^{n-1}

ddx(x2)+2y1ddx(y)=0\frac{d}{dx}\left(x^2\right)+2y^{1}\frac{d}{dx}\left(y\right)=0
6

Any expression to the power of 11 is equal to that same expression

ddx(x2)+2yddx(y)=0\frac{d}{dx}\left(x^2\right)+2y\frac{d}{dx}\left(y\right)=0
7

The derivative of the linear function is equal to 11

ddx(x2)+2yβ‹…yβ€²=0\frac{d}{dx}\left(x^2\right)+2y\cdot y^{\prime}=0

The power rule for differentiation states that if nn is a real number and f(x)=xnf(x) = x^n, then fβ€²(x)=nxnβˆ’1f'(x) = nx^{n-1}

2x(2βˆ’1)2x^{\left(2-1\right)}

Subtract the values 22 and βˆ’1-1

2x2x
8

The power rule for differentiation states that if nn is a real number and f(x)=xnf(x) = x^n, then fβ€²(x)=nxnβˆ’1f'(x) = nx^{n-1}

2x+2yβ‹…yβ€²=02x+2y\cdot y^{\prime}=0
9

We need to isolate the dependent variable yy, we can do that by simultaneously subtracting 2x2x from both sides of the equation

2yβ‹…yβ€²=βˆ’2x2y\cdot y^{\prime}=-2x
10

Divide both sides of the equation by 22

yβ€²y=βˆ’2x2y^{\prime}y=\frac{-2x}{2}
11

Take βˆ’22\frac{-2}{2} out of the fraction

yβ€²y=βˆ’xy^{\prime}y=-x
12

Divide both sides of the equation by yy

yβ€²=βˆ’xyy^{\prime}=\frac{-x}{y}

ξ ƒ Risposta finale al problema

yβ€²=βˆ’xyy^{\prime}=\frac{-x}{y} ξ ƒ

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