Find the derivative of y with respect to z. y=e(z9) dydz=

Question

Find the derivative of y with respect to z.  y=e(z9)  dydz=

Elberte · Accepted Answer

Step 1
We have to find the derivative of y with respect to z when

y = e^{(z^{9})}

Taking log both sides,

\ln (y) = {\ln 3}^{(z^{9})}

= z^{9} \ln (3)

...(1)
Here we have used following property:

\ln (a^{m}) = m \ln (a)

We know the formula of derivatives:

\frac{{d x}^{n}}{d x} = n x^{n - 1}

\frac{d \ln (x)}{d x} = \frac{1}{x}

\frac{d \ln (f (x))}{d x} = \frac{1}{f (x)} \frac{d f (x)}{d x}

Step 2
Differentiating equation (1) with respect to z,

\frac{d \ln (y)}{d z} = \ln (3) \frac{{d z}^{9}}{d z}

\frac{1}{y} \frac{d y}{d z} = \ln (3) (9 z^{9 - 1})

\frac{d y}{d z} = 9 y \ln (3) z^{8}

Putting value of y,

\frac{d y}{d z} = 9 (3^{z^{9}}) \ln (3) z^{8}

Hence, derivative of y with respect to z is

9 (3^{z^{9}}) \ln (3) z^{8}

.

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

Find the derivative of y with respect to z. y=e^{(z^{9})}