What is the derivative of {f}(x(e^x)) ?

The derivative of {f}(x(e^x)) is derivative of {f}(x)e^x+e^x{f}(x)

What is the first derivative of {f}(x(e^x)) ?

The first derivative of {f}(x(e^x)) is derivative of {f}(x)e^x+e^x{f}(x)

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\frac{d}{d x} (f (x) (e^{x}))

\frac{d}{d x} (f (x)) e^{x} + e^{x} f (x)

Solution steps

\frac{d}{d x} (f (x) (e^{x}))

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = f (x), g = e^{x}

= \frac{d}{d x} (f (x)) e^{x} + \frac{d}{d x} (e^{x}) f (x)

\frac{d}{d x} (e^{x}) = e^{x}

= \frac{d}{d x} (f (x)) e^{x} + e^{x} f (x)

What is the derivative of {f}(x(e^x)) ?
The derivative of {f}(x(e^x)) is derivative of {f}(x)e^x+e^x{f}(x)
What is the first derivative of {f}(x(e^x)) ?
The first derivative of {f}(x(e^x)) is derivative of {f}(x)e^x+e^x{f}(x)