What is (\partial)/(\partial x)(e^{(z+e^x+1)}) ?

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

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(\partial)/(\partial x)(e^{(z+e^x+1)})

\frac{\partial}{\partial x} (e^{(z + e^{x} + 1)})

e^{z + e^{x} + 1 + x}

Solution steps

\frac{\partial}{\partial x} (e^{(z + e^{x} + 1)})

Treat

z

as a constant

Apply the chain rule:

e^{z + e^{x} + 1} \frac{\partial}{\partial x} (z + e^{x} + 1)

= e^{z + e^{x} + 1} \frac{\partial}{\partial x} (z + e^{x} + 1)

\frac{\partial}{\partial x} (z + e^{x} + 1) = e^{x}

= e^{z + e^{x} + 1} e^{x}

Apply exponent rule:

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

e^{z + e^{x} + 1} e^{x} = e^{z + e^{x} + 1 + x}

= e^{z + e^{x} + 1 + x}

\frac{\partial}{\partial x} (- 3 cos (3 x))

\int_{6}^{8} \frac{80}{( x - 6 ) ^{3}} d x

\frac{\partial}{\partial y} (x^{2} - y^{3} + 5 x + 12 y + 1)

2 x^{2} + 3

\int (e^{7 x}) d x

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