What is the derivative of y^{3x} ?

The derivative of y^{3x} is 3(ln(y)+(x(dy)/(dx))/y)y^{3x}

What is the first derivative of y^{3x} ?

The first derivative of y^{3x} is 3(ln(y)+(x(dy)/(dx))/y)y^{3x}

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\frac{d}{d x} (y^{3 x})

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

Solution steps

\frac{d}{d x} (y^{3 x})

Apply exponent rule:

a^{b} = e^{b l n (a)}

y^{3 x} = e^{3 x l n (y)}

= \frac{d}{d x} (e^{3 x l n (y)})

Apply the chain rule:

e^{3 x l n (y)} \frac{d}{d x} (3 x ln (y))

= e^{3 x l n (y)} \frac{d}{d x} (3 x ln (y))

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

= e^{3 x l n (y)} \cdot 3 (ln (y) + \frac{x \frac{d y}{d x}}{y})

e^{3 x l n (y)} = y^{3 x}

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

What is the derivative of y^{3x} ?
The derivative of y^{3x} is 3(ln(y)+(x(dy)/(dx))/y)y^{3x}
What is the first derivative of y^{3x} ?
The first derivative of y^{3x} is 3(ln(y)+(x(dy)/(dx))/y)y^{3x}