Question

How to differentiate `y=x^tanx` with respect to x?

Answer 1

Use logarithmic differentiation

Explanation:

The procedure for logarithmic differentiation is as follows.

Take the natural logarithm of both sides of the equation:

`ln(y) = ln(x^tan(x))`

Use the properties of logarithms to simplify the right side into a form that can be differentiated. In this case, we shall use the property `ln(a^c) = cln(a)`:

`ln(y) = tan(x)ln(x)`

Please observe that the right side is, now, a product. We know that how to differentiate a product, therefore, we differentiate both sides:

`(d(ln(y)))/dx = (d(tan(x)ln(x)))/dx`

Use the product rule, `(d(uv))/dx = (du)/dxv + u(dv)/dx` on the right side where `u=tan(x), v = ln(x), (du)/dx= sec^2(x), and (dv)/dx = 1/x`:

`(d(ln(y)))/dx = sec^2(x)ln(x) + tan(x)/x`

Use the chain rule, `(d(f(y)))/dx = (d(f(y)))/dydy/dx` on the left side where `f(y) = ln(y) and (d(f(y)))/dy = 1/y`:

`1/ydy/dx = sec^2(x)ln(x) + tan(x)/x`

If you have done the logarithimic differentiation correctly, the "next-to-last" step must always be, multiply both sides by y:

`dy/dx = (sec^2(x)ln(x) + tan(x)/x)y`

And the last step must always be a substitution for the equation for y. In this case, it is, `y=x^tanx`:

`dy/dx = (sec^2(x)ln(x) + tan(x)/x)x^tan(x)`