What is (\partial)/(\partial x)(e^{4(e^x-1)}) ?

The answer to (\partial)/(\partial x)(e^{4(e^x-1)}) is 4e^{4(e^x-1)+x}

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(\partial)/(\partial x)(e^{4(e^x-1)})

\frac{\partial}{\partial x} (e^{4 (e^{x} - 1)})

4 e^{4 (e^{x} - 1) + x}

Solution steps

\frac{\partial}{\partial x} (e^{4 (e^{x} - 1)})

Apply the chain rule:

e^{4 (e^{x} - 1)} \frac{\partial}{\partial x} (4 (e^{x} - 1))

= e^{4 (e^{x} - 1)} \frac{\partial}{\partial x} (4 (e^{x} - 1))

\frac{\partial}{\partial x} (4 (e^{x} - 1)) = 4 e^{x}

= e^{4 (e^{x} - 1)} \cdot 4 e^{x}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

e^{4 (e^{x} - 1)} e^{x} = e^{4 (e^{x} - 1) + x}

= 4 e^{4 (e^{x} - 1) + x}

What is (\partial)/(\partial x)(e^{4(e^x-1)}) ?
The answer to (\partial)/(\partial x)(e^{4(e^x-1)}) is 4e^{4(e^x-1)+x}