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ki77a [65]
2 years ago
8

Find y' if x^y = y^x​

Mathematics
1 answer:
Brrunno [24]2 years ago
5 0

The answer is \boxed {\frac{dy}{dx} = \frac{y^{x}logy-yx^{y-1}}{x^{y}logx-xy^{x-1}}}.

Apply logarithmic differentiation on each side.

<u>LHS</u>

  • u = x^y
  • log u = y log x
  • 1/u du/dx = y/x + log x (dy/dx)
  • du/dx = x^y (y/x + log x dy/dx)
  • du/dx = yx^(y - 1) + x^y logx dy/dx

<u>RHS</u>

  • v = y^x
  • log v = x log y
  • 1/v dv/dx = x/y dy/dx + logy
  • dv/dx = y^x (x/y dy/dx + logy)
  • dv/dx = xy^(x - 1) dy/dx + y^x logy

<u>Equating both sides</u>

  • yx^(y - 1) + x^y logx dy/dx = xy^(x - 1) dy/dx + y^x logy
  • dy/dx (x^y logx - xy^(x - 1)) = y^x logy - yx^(y - 1)
  • \boxed {\frac{dy}{dx} = \frac{y^{x}logy-yx^{y-1}}{x^{y}logx-xy^{x-1}}}
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Solve the triangle.
Reika [66]

Answer:

cute scalene triangle.

Sides: a = 1   b = 2   c = 1.906

Area: T = 0.94

Perimeter: p = 4.906

Semiperimeter: s = 2.453

Angle ∠ A = α = 29.543° = 29°32'36″ = 0.516 rad

Angle ∠ B = β = 80.457° = 80°27'24″ = 1.404 rad

Angle ∠ C = γ = 70° = 1.222 rad

Height: ha = 1.879

Height: hb = 0.94

Height: hc = 0.986

Median: ma = 1.888

Median: mb = 1.147

Median: mc = 1.262

Inradius: r = 0.383

Circumradius: R = 1.014

Vertex coordinates: A[1.906; 0] B[0; 0] C[0.166; 0.986]

Centroid: CG[0.691; 0.329]

Coordinates of the circumscribed circle: U[0.953; 0.347]

Coordinates of the inscribed circle: I[0.453; 0.383]

Exterior (or external, outer) angles of the triangle:

∠ A' = α' = 150.457° = 150°27'24″ = 0.516 rad

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∠ C' = γ' = 110° = 1.222 rad

Step-by-step explanation:

7 0
3 years ago
The distance between two cities is 180km. The distance was covered by a car at the rate of 50km/h, by a bus 45km/h, by a truck a
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Answer:

The speed and time are directly proportional in that the higher the speed the lesser the time and vice versa.

The constant is the total distance which is 180 km

Step-by-step explanation:

Step one:

given data

The distance between the  two cities is 180km

1.The distance was covered by a car at the rate of 50km/h

The time taken will be

velocity= distance/time

time= distance/velocity

time= 180/50

time= 3.6 hours

2.The distance was covered by a bus 45km/h

The time taken will be

velocity= distance/time

time= distance/velocity

time= 180/45

time= 4 hours

3.The distance was covered by a bicycle at 25km/h

The time taken will be

velocity= distance/time

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3 years ago
Please help!! Local wildlife experts have begun to track the population of trout in their area. The function f represents the ap
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Answer:

Option C.

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The total trout population in the area is described as

f(t) + g(t)

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Answer:

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