How do you differentiate the following

Question

f(v) = [ln(1 + v ^ 2)] ^ e

f(x) = (cos(x) + 3) ^ tan(x)

Answer 1

For f(x)=(cosx +3)^tanx, let y = f(x) and take the natural log of both sides:

y = (cosx + 3)^tanx

ln y = ln [(cosx + 3)^tanx]

ln y = tanx ⋅ ln(cosx+3)

Now take the derivative of both sides using the product rule on the right:

1/y (y') = tanx [1/(cosx + 3)](-sinx) + ln(cosx+3) ⋅ sec²(x)

Simplify the right side a bit and multiply both sides by y:

y' = (cosx+3)^tanx [-sin²x/(cos²x+ 3cosx) + sec²(x)ln(cosx + 3)]

This graph shows tangent lines to the original function at a sliding point of tangency (using the above derivative to determine the slope of the tangent line):

desmos.com/calculator/1zgdvy8qbh