Find the Derivative - d/da tan( natural log of ax+b)

Problem

d(tan(ln(a*x+b)))/d(a)

Solution

1. Identify the variable of differentiation, which is a The terms x and b are treated as constants.

2. Apply the chain rule for the outermost function, which is tan(u) The derivative of tan(u) is sec^2(u)

d(tan(ln(a*x+b)))/d(a)=sec^2(ln(a*x+b))⋅d(ln(a*x+b))/d(a)

3. Apply the chain rule for the natural logarithm function. The derivative of ln(u) is 1/u⋅d(u)/d(a)

d(ln(a*x+b))/d(a)=1/(a*x+b)⋅d(a*x+b)/d(a)

4. Differentiate the innermost expression a*x+b with respect to a Since x and b are constants, the derivative is x

d(a*x+b)/d(a)=x

5. Combine all the components obtained from the chain rule steps.

d(tan(ln(a*x+b)))/d(a)=sec^2(ln(a*x+b))⋅1/(a*x+b)⋅x

6. Simplify the expression into a single fraction.

d(tan(ln(a*x+b)))/d(a)=(x*sec^2(ln(a*x+b)))/(a*x+b)

Final Answer

d(tan(ln(a*x+b)))/d(a)=(x*sec^2(ln(a*x+b)))/(a*x+b)

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