Calculus homework question help please

Question

Use logarithmic differentiation to find the derivative of 𝑦=(sin(4𝑥))^(1+3𝑥):

𝑦′= 

Answer 1

The last step would be to replace the y on the right with "(sin(4x))^(1+3x)" per the original definition of "y".

Answer 2

Take the natural log of both sides so that you can use properties of logarithms as follows:

ln(y) = ln[sin(4x)^(3x+1)]

ln(y) = (3x+1)ln[sin(4x)] (power property of logarithms gets you to here)

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

y'/y = (3x+1)[1/sin(4x)][cos(4x)(4)] + 3ln[sin(4x)]

y'/y = 4(3x+1)cot(4x) + 3ln[sin(4x)]

Multiply both sides by y to isolate y'.

y' = (you can complete this).

Here is a Desmos graph where the original function has been restricted to the domain 0 to pi/4. This graph shows tangent lines to the graph of the original function on that interval:

desmos.com/calculator/cjfmouy4z3