Pergunta :
y z - ln{ z } = x + y
Solve the equation
Solve for x
Solve for y
x=yz−ln(z)−y
Evaluate
yz−ln(z)=x+y
Swap the sides of the equation
x+y=yz−ln(z)
Solução
x=yz−ln(z)−y
Mostrar solução
Find the partial derivative
Find ∂x∂z by differentiating the equation directly
Find ∂y∂z by differentiating the equation directly
∂x∂z=yz−1z
Evaluate
yz−ln(z)=x+y
Find ∂x∂z by taking the derivative of both sides with respect to x
∂x∂(yz−ln(z))=∂x∂(x+y)
Use differentiation rule ∂x∂(f(x)±g(x))=∂x∂(f(x))±∂x∂(g(x))
∂x∂(yz)−∂x∂(ln(z))=∂x∂(x+y)
Evaluate
Mais Passos
Evaluate
∂x∂(yz)
Use differentiation rule ∂x∂(cf(x))=c×∂x∂(f(x))
y×∂x∂(z)
Find the derivative
y∂x∂z
y∂x∂z−∂x∂(ln(z))=∂x∂(x+y)
Evaluate
Mais Passos
Evaluate
∂x∂(ln(z))
Use the chain rule ∂x∂(f(g))=∂g∂(f(g))×∂x∂(g) where the g=z, to find the derivative
∂g∂(ln(g))×∂x∂(z)
Use ∂x∂lnx=x1 to find derivative
g1×∂x∂(z)
Evaluate
g1×∂x∂z
Substitute back
z1×∂x∂z
Multiply the terms
z∂x∂z
y∂x∂z−z∂x∂z=∂x∂(x+y)
Calculate
Mais Passos
Evaluate
y∂x∂z−z∂x∂z
Reduce fractions to a common denominator
zy∂x∂z×z−z∂x∂z
Write all numerators above the common denominator
zy∂x∂z×z−∂x∂z
Use the commutative property to reorder the terms
zyz∂x∂z−∂x∂z
zyz∂x∂z−∂x∂z=∂x∂(x+y)
Use differentiation rule ∂x∂(f(x)±g(x))=∂x∂(f(x))±∂x∂(g(x))
zyz∂x∂z−∂x∂z=∂x∂(x)+∂x∂(y)
Use ∂x∂xn=nxn−1 to find derivative
zyz∂x∂z−∂x∂z=1+∂x∂(y)
Use ∂x∂(c)=0 to find derivative
zyz∂x∂z−∂x∂z=1+0
Removing 0 doesn't change the value,so remove it from the expression
zyz∂x∂z−∂x∂z=1
Cross multiply
yz∂x∂z−∂x∂z=z
Collect like terms by calculating the sum or difference of their coefficients
(yz−1)∂x∂z=z
Divide both sides
yz−1(yz−1)∂x∂z=yz−1z
Solução
∂x∂z=yz−1z
Mostrar solução
