# How do you differentiate #y=b^x?

God bless.....I hope the explanation is useful.

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To differentiate ( y = b^x ), where ( b ) is a constant base and ( x ) is the variable exponent, we use logarithmic differentiation. The steps are as follows:

- Take the natural logarithm of both sides: ( \ln(y) = \ln(b^x) ).
- Apply the logarithmic property ( \ln(b^x) = x \ln(b) ).
- Differentiate both sides with respect to ( x ).
- Use the chain rule when differentiating ( \ln(y) ).
- Solve for ( \frac{{dy}}{{dx}} ).

The result will be:

[ \frac{{dy}}{{dx}} = \frac{{d}}{{dx}}(b^x) = b^x \cdot \ln(b) ]

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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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