# What is the second derivative for #y(x)=5e^(pi x)+cos (y(x))# ?

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To find the second derivative of the function ( y(x) = 5e^{\pi x} + \cos(y(x)) ), you first need to find the first derivative with respect to ( x ), then take the derivative of that result again with respect to ( x ).

First derivative: [ y'(x) = 5\pi e^{\pi x} - \sin(y(x)) \cdot y'(x) ]

Solve for ( y'(x) ): [ y'(x) + \sin(y(x)) \cdot y'(x) = 5\pi e^{\pi x} ] [ y'(x) (1 + \sin(y(x))) = 5\pi e^{\pi x} ] [ y'(x) = \frac{5\pi e^{\pi x}}{1 + \sin(y(x))} ]

Now, take the derivative of ( y'(x) ) with respect to ( x ) to find the second derivative:

[ \frac{d^2y}{dx^2} = \frac{d}{dx} \left( \frac{5\pi e^{\pi x}}{1 + \sin(y(x))} \right) ]

Using the quotient rule:

[ \frac{d^2y}{dx^2} = \frac{(5\pi e^{\pi x})'(1 + \sin(y(x))) - 5\pi e^{\pi x}(1 + \sin(y(x)))'}{(1 + \sin(y(x)))^2} ]

[ = \frac{(5\pi^2 e^{\pi x})(1 + \sin(y(x))) - 5\pi e^{\pi x} \cos(y(x)) \cdot y'(x)}{(1 + \sin(y(x)))^2} ]

Substitute ( y'(x) ) into the equation:

[ = \frac{(5\pi^2 e^{\pi x})(1 + \sin(y(x))) - 5\pi e^{\pi x} \cos(y(x)) \cdot \frac{5\pi e^{\pi x}}{1 + \sin(y(x))}}{(1 + \sin(y(x)))^2} ]

[ = \frac{(5\pi^2 e^{\pi x})(1 + \sin(y(x))) - (25\pi^2 e^{2\pi x}) \cos(y(x))}{(1 + \sin(y(x)))^2} ]

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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