What is derivative of x^y=x^3+2sin(x) ?

The answer to derivative of x^y=x^3+2sin(x) is (dy)/(dx)=(x^{1-y}(3x^2+2cos(x))-y)/(xln(x))

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derivative of x^y=x^3+2sin(x)

derivative of

x^{y} = x^{3} + 2 sin (x)

\frac{d y}{d x} = \frac{x ^{1 - y} ( 3 x ^{2} + 2 cos ( x ) ) - y}{x ln ( x )}

Solution steps

x^{y} = x^{3} + 2 sin (x)

Treat

y

y (x)

Differentiate both sides:

x^{y} (ln (x) \frac{d y}{d x} + \frac{y}{x}) = 3 x^{2} + 2 cos (x)

Isolate

\frac{d y}{d x} : \frac{d y}{d x} = \frac{x ^{1 - y} ( 3 x ^{2} + 2 cos ( x ) ) - y}{x ln ( x )}

\frac{d y}{d x} = \frac{x ^{1 - y} ( 3 x ^{2} + 2 cos ( x ) ) - y}{x ln ( x )}

1 - x^{3} = t^{2}

(x + y)^{2} = 4

x^{2} - y^{2} = tan (x) y

x = \frac{y + 1}{y}

2 x^{3} + 4 x^{2} - y = x^{6}

What is derivative of x^y=x^3+2sin(x) ?
The answer to derivative of x^y=x^3+2sin(x) is (dy)/(dx)=(x^{1-y}(3x^2+2cos(x))-y)/(xln(x))