Find the Derivative - d/dx sin(tan(3x))

Problem

d(sin(tan(3*x)))/d(x)

Solution

1. Identify the outer function as sin(u) where u=tan(3*x)

2. Apply the chain rule by differentiating the outer function with respect to its inner argument.

d(sin(u))/d(u)=cos(u)

3. Differentiate the inner function u=tan(3*x) with respect to x which requires a second application of the chain rule.

d(tan(3*x))/d(x)=sec2(3*x)⋅(d(3)*x)/d(x)

4. Calculate the derivative of the innermost linear term 3*x

(d(3)*x)/d(x)=3

5. Combine all parts using the chain rule formula.

d(sin(tan(3*x)))/d(x)=cos(tan(3*x))⋅sec2(3*x)⋅3

6. Simplify the expression by moving the constant to the front.

3*sec2(3*x)*cos(tan(3*x))

Final Answer

d(sin(tan(3*x)))/d(x)=3*sec2(3*x)*cos(tan(3*x))

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