Find the Derivative - d/dx y=sin(tan(4x))

Problem

d()/d(x)*sin(tan(4*x))

Solution

1. Identify the outer, middle, and inner functions to apply the chain rule. The outer function is sin(u) the middle function is tan(v) and the inner function is 4*x

2. Differentiate the outer function sin(u) with respect to its argument, which results in cos(u)

3. Differentiate the middle function tan(v) with respect to its argument, which results in sec2(v)

4. Differentiate the inner function 4*x with respect to x which results in 4

5. Multiply the derivatives together according to the chain rule formula d(y)/d(x)=d(y)/d(u)⋅d(u)/d(v)⋅d(v)/d(x)

d(sin(tan(4*x)))/d(x)=cos(tan(4*x))⋅d(tan(4*x))/d(x)

d(sin(tan(4*x)))/d(x)=cos(tan(4*x))⋅sec2(4*x)⋅(d(4)*x)/d(x)

d(sin(tan(4*x)))/d(x)=cos(tan(4*x))⋅sec2(4*x)⋅4

6. Rearrange the terms to write the final derivative clearly.

Final Answer

d(sin(tan(4*x)))/d(x)=4*sec2(4*x)*cos(tan(4*x))

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