How do you differentiate #f(x)=tan(e^x) # using the chain rule?
Using the chain rule:
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To differentiate ( f(x) = \tan(e^x) ) using the chain rule, follow these steps:
- Identify the outer function ( \tan(u) ) and the inner function ( u = e^x ).
- Find the derivative of the inner function ( u = e^x ) with respect to ( x ), which is ( \frac{du}{dx} = e^x ).
- Compute the derivative of the outer function ( \tan(u) ) with respect to ( u ), which is ( \sec^2(u) ).
- Apply the chain rule by multiplying the derivatives obtained in steps 2 and 3: ( \frac{df}{dx} = \frac{du}{dx} \cdot \frac{d}{du}(\tan(u)) = e^x \cdot \sec^2(u) ).
- Substitute ( u = e^x ) back into the expression to get the final derivative: ( \frac{df}{dx} = e^x \cdot \sec^2(e^x) ).
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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.
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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