Question
Function
Find the first partial derivative with respect to u
Find the first partial derivative with respect to m
∂u∂v=36u2m4−36m5u2
Evaluate
v×1=(m×1−m2)u×12m2×mu2×1×m2
Any expression multiplied by 1 remains the same
v=(m×1−m2)u×12m2×mu2×1×m2
Simplify
More Steps

Evaluate
(m×1−m2)u×12m2×mu2×1×m2
Any expression multiplied by 1 remains the same
(m−m2)u×12m2×mu2×1×m2
Rewrite the expression
(m−m2)u×12m2×mu2×m2
Multiply the terms with the same base by adding their exponents
(m−m2)u×12m2+2×mu2
Add the numbers
(m−m2)u×12m4×mu2
Multiply the terms
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Evaluate
u×12m4×mu2
Use the commutative property to reorder the terms
12um4×mu2
Cancel out the common factor m
12um3u2
Multiply the terms
12u3m3
(m−m2)×12u3m3
Multiply the terms
12u3m3(m−m2)
v=12u3m3(m−m2)
Find the first partial derivative by treating the variable m as a constant and differentiating with respect to u
∂u∂v=∂u∂(12u3m3(m−m2))
Use differentiation rule ∂x∂(f(x)×g(x))=∂x∂(f(x))×g(x)+f(x)×∂x∂(g(x))
∂u∂v=∂u∂(12)u3m3(m−m2)+12×∂u∂(u3m3)(m−m2)+12u3m3×∂u∂(m−m2)
Evaluate
∂u∂v=0×u3m3(m−m2)+12×∂u∂(u3m3)(m−m2)+12u3m3×∂u∂(m−m2)
Evaluate
∂u∂v=0+12×∂u∂(u3m3)(m−m2)+12u3m3×∂u∂(m−m2)
Evaluate
More Steps

Evaluate
∂u∂(u3m3)
Use differentiation rule ∂x∂(cf(x))=c×∂x∂(f(x))
m3×∂u∂(u3)
Use ∂x∂xn=nxn−1 to find derivative
m3×3u2
Multiply the terms
3u2m3
∂u∂v=0+12×3u2m3(m−m2)+12u3m3×∂u∂(m−m2)
Evaluate
∂u∂v=0+36u2m4−36m5u2+12u3m3×∂u∂(m−m2)
Use ∂x∂(c)=0 to find derivative
∂u∂v=0+36u2m4−36m5u2+12u3m3×0
Evaluate
∂u∂v=0+36u2m4−36m5u2+0
Solution
∂u∂v=36u2m4−36m5u2
Show Solution

Solve the equation
Solve for u
Solve for v
u=6m2−6m3318m2v−36m3v+18m4v
Evaluate
v×1=(m×1−m2)u×12m2×mu2×1×m2
Any expression multiplied by 1 remains the same
v=(m×1−m2)u×12m2×mu2×1×m2
Simplify
More Steps

Evaluate
(m×1−m2)u×12m2×mu2×1×m2
Any expression multiplied by 1 remains the same
(m−m2)u×12m2×mu2×1×m2
Rewrite the expression
(m−m2)u×12m2×mu2×m2
Multiply the terms with the same base by adding their exponents
(m−m2)u×12m2+2×mu2
Add the numbers
(m−m2)u×12m4×mu2
Multiply the terms
More Steps

Evaluate
u×12m4×mu2
Use the commutative property to reorder the terms
12um4×mu2
Cancel out the common factor m
12um3u2
Multiply the terms
12u3m3
(m−m2)×12u3m3
Multiply the terms
12u3m3(m−m2)
v=12u3m3(m−m2)
Rewrite the expression
v=(12m4−12m5)u3
Swap the sides of the equation
(12m4−12m5)u3=v
Divide both sides
12m4−12m5(12m4−12m5)u3=12m4−12m5v
Divide the numbers
u3=12m4−12m5v
Take the 3-th root on both sides of the equation
3u3=312m4−12m5v
Calculate
u=312m4−12m5v
Simplify the root
More Steps

Evaluate
312m4−12m5v
To take a root of a fraction,take the root of the numerator and denominator separately
312m4−12m53v
Simplify the radical expression
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Evaluate
312m4−12m5
Factor the expression
312m4(1−m)
The root of a product is equal to the product of the roots of each factor
312×3m4×31−m
Reduce the index of the radical and exponent with 3
312×m3m×31−m
Calculate the product
m312m×31−m
Calculate the product
m312m−12m2
m312m−12m23v
Multiply by the Conjugate
m312m−12m2×3(12m−12m2)23v×3(12m−12m2)2
Calculate
m(12m−12m2)3v×3(12m−12m2)2
Calculate
More Steps

Evaluate
3v×3(12m−12m2)2
The product of roots with the same index is equal to the root of the product
3v(12m−12m2)2
Calculate the product
3144v(m−m2)2
m(12m−12m2)3144v(m−m2)2
Calculate
12m2−12m33144v(m−m2)2
u=12m2−12m33144v(m−m2)2
Solution
More Steps

Evaluate
12m2−12m33144v(m−m2)2
Rewrite the expression
More Steps

Evaluate
3144v(m−m2)2
Rewrite the expression
3144×3v×3(m−m2)2
Simplify the root
2318(m−m2)2v
12m2−12m32318(m−m2)2v
Reduce the fraction
6m2−6m3318(m−m2)2v
Expand the expression
More Steps

Evaluate
18(m−m2)2v
Calculate
18(m2−2m3+m4)v
Calculate
18m2v−36m3v+18m4v
6m2−6m3318m2v−36m3v+18m4v
u=6m2−6m3318m2v−36m3v+18m4v
Show Solution
