# How do you differentiate #f(t)=tan(e^t)+e^(tant)#?

Use the chain rule.

Using the chain rule, we first take the derivative of the "outside" term, then multiply the result by the derivative of the "inside" term.

We now simply add the derivatives, as specified by our original function. Our final answer is:

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To differentiate ( f(t) = \tan(e^t) + e^{\tan(t)} ), apply the chain rule and the derivative of the tangent function.

The derivative of ( \tan(x) ) is ( \sec^2(x) ).

The derivative of ( e^x ) is ( e^x ).

Using the chain rule, the derivative of ( \tan(e^t) ) is ( \sec^2(e^t) \cdot e^t ).

The derivative of ( e^{\tan(t)} ) is ( e^{\tan(t)} \cdot \sec^2(t) ).

Therefore, the derivative of ( f(t) ) is ( \frac{d}{dt}( \tan(e^t) + e^{\tan(t)}) = \sec^2(e^t) \cdot e^t + e^{\tan(t)} \cdot \sec^2(t) ).

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