What is the derivative of e^x(cos(x-sin(x))) ?

The derivative of e^x(cos(x-sin(x))) is -2e^xsin(x)

What is the first derivative of e^x(cos(x-sin(x))) ?

The first derivative of e^x(cos(x-sin(x))) is -2e^xsin(x)

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\frac{d}{d x} (e^{x} (cos (x) - sin (x)))

- 2 e^{x} sin (x)

Solution steps

\frac{d}{d x} (e^{x} (cos (x) - sin (x)))

Apply the Product Rule:

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

f = e^{x}, g = cos (x) - sin (x)

= \frac{d}{d x} (e^{x}) (cos (x) - sin (x)) + \frac{d}{d x} (cos (x) - sin (x)) e^{x}

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

\frac{d}{d x} (cos (x) - sin (x)) = - sin (x) - cos (x)

= e^{x} (cos (x) - sin (x)) + (- sin (x) - cos (x)) e^{x}

Simplify

e^{x} (cos (x) - sin (x)) + (- sin (x) - cos (x)) e^{x} : - 2 e^{x} sin (x)

= - 2 e^{x} sin (x)

What is the derivative of e^x(cos(x-sin(x))) ?
The derivative of e^x(cos(x-sin(x))) is -2e^xsin(x)
What is the first derivative of e^x(cos(x-sin(x))) ?
The first derivative of e^x(cos(x-sin(x))) is -2e^xsin(x)