Find dydx if y=log9(x2−1) .

Answer 1

To find dy/dx, we need to use the chain rule. The chain rule states that if we have a composite function y = f(g(x)), then the derivative dy/dx can be found by multiplying the derivative of the outer function f'(g(x)) with the derivative of the inner function g'(x).

In this case, the outer function is y = log9(u), where u = x^2 - 1, and the inner function is u = x^2 - 1.

Let's calculate the derivatives:

The derivative of the outer function y = log9(u) is dy/du = 1/(u ln 9), because the derivative of log9(u) is 1/(u ln 9).

The derivative of the inner function u = x^2 - 1 is du/dx = 2x, because the derivative of x^2 - 1 is 2x.

Now, we can multiply the derivatives to find dy/dx:

dy/dx = (dy/du)(du/dx)
= (1/(u ln 9))(2x)
= 2x/(u ln 9)
= 2x/((x^2 - 1) ln 9)

Therefore, dy/dx = 2x/((x^2 - 1) ln 9).