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Question

How to use logarithmic differentiation to determine y′ if cos x = e^y.

Answer 1

To use logarithmic differentiation to find y′ when cos x = e^y, we can follow these steps:

1. Take the natural logarithm of both sides of the equation:
2. ln(cos x) = ln(e^y)
3. Use the logarithmic rule that ln(a^b) = b ln(a) to simplify the right-hand side:
4. ln(cos x) = y ln(e)
5. ln(cos x) = y
6. Differentiate both sides of the equation with respect to x:
7. d/dx [ln(cos x)] = d/dx [y]
8. -sin x/cos x = dy/dx
9. Simplify the left-hand side using the trigonometric identity tan x = sin x/cos x:
10. -sin x = dy/dx

Therefore, we have found that y′ = -sin x.