Solve 2^x^2^y= 2^x y

Solve the equation

y = 1

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Testing for symmetry about the origin

Testing for symmetry about the x-axis

Testing for symmetry about the y-axis

Not symmetry with respect to the origin

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Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = ln (2) - y ln (2)

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Find the second derivative with respect to x

Find the second derivative with respect to y

\frac{d ^{2} y}{d x ^{2}} = - (ln (2))^{2} + (ln (2))^{2} y

Calculate

2^{x} = 2^{x} y

Take the derivative of both sides

\frac{d}{d x} (2^{x}) = \frac{d}{d x} (2^{x} y)

Calculate the derivative

ln (2) \times 2^{x} = \frac{d}{d x} (2^{x} y)

Calculate the derivative

More Steps

ln (2) \times 2^{x} = ln (2) \times 2^{x} y + 2^{x} \frac{d y}{d x}

Rewrite the expression

2^{x} ln (2) = 2^{x} y ln (2) + 2^{x} \frac{d y}{d x}

Swap the sides of the equation

2^{x} y ln (2) + 2^{x} \frac{d y}{d x} = 2^{x} ln (2)

Move the expression to the right-hand side and change its sign

2^{x} \frac{d y}{d x} = 2^{x} ln (2) - 2^{x} y ln (2)

Divide both sides

\frac{2 ^{x} \frac{d y}{d x}}{2 ^{x}} = \frac{2 ^{x} ln ( 2 ) - 2 ^{x} y ln ( 2 )}{2 ^{x}}

Divide the numbers

\frac{d y}{d x} = \frac{2 ^{x} ln ( 2 ) - 2 ^{x} y ln ( 2 )}{2 ^{x}}

Divide the numbers

More Steps

\frac{d y}{d x} = ln (2) - y ln (2)

Rewrite the expression

\frac{d y}{d x} = ln (2) - ln (2) \times y

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (ln (2) - ln (2) \times y)

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (ln (2) - ln (2) \times y)

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (ln (2)) + \frac{d}{d x} (- ln (2) \times y)

Use \frac{d}{d x} (c) = 0 to find derivative

\frac{d ^{2} y}{d x ^{2}} = 0 + \frac{d}{d x} (- ln (2) \times y)

Evaluate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = 0 - ln (2) \times \frac{d y}{d x}

Evaluate

\frac{d ^{2} y}{d x ^{2}} = - ln (2) \times \frac{d y}{d x}

Use equation \frac{d y}{d x} = ln (2) - ln (2) \times y to substitute

\frac{d ^{2} y}{d x ^{2}} = - ln (2) \times (ln (2) - ln (2) \times y)

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = - (ln (2))^{2} + (ln (2))^{2} y

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