Question

Use logarithmic differentiation to find the derivative of the function y = (natural log (x + 1)) raised to the power of cosine (-3x).

Answer 1

To find the derivative of the function y = ln(x + 1)^cos(-3x), we can use logarithmic differentiation. The derivative is y' = y * 3 * sin(-3x) * ln(ln(x + 1)).

Step-by-step explanation:

To find the derivative of the function y = ln(x + 1)cos(-3x), we can use logarithmic differentiation. Here are the steps:

1. Take the natural logarithm of both sides of the equation to simplify the expression: ln(y) = ln(ln(x + 1)cos(-3x)).
2. Apply the logarithmic property: ln(ab) = b * ln(a). This gives us: ln(y) = cos(-3x) * ln(ln(x + 1)).
3. Differentiate both sides with respect to x. The derivative of ln(y) with respect to x is (1/y) * y', and the derivative of cos(-3x) is 3 * sin(-3x). We then have: (1/y) * y' = 3 * sin(-3x) * ln(ln(x + 1)).
4. Multiply both sides by y to isolate y': y' = y * 3 * sin(-3x) * ln(ln(x + 1)).

Therefore, the derivative of y = ln(x + 1)cos(-3x) is y' = y * 3 * sin(-3x) * ln(ln(x + 1)).