Find the Derivative - d/dx tan(x^4+1)^2

Problem

d(tan(x^4+1))/d(x)

Solution

1. Identify the outer function as the square power, requiring the power rule combined with the chain rule.

2. Apply the power rule to the expression tan(x^4+1) which brings the exponent to the front and reduces it by one.

2*tan(x^4+1)⋅d(tan(x^4+1))/d(x)

3. Apply the chain rule to the derivative of the tangent function, where the derivative of tan(u) is sec^2(u)

2*tan(x^4+1)⋅sec^2(x^4+1)⋅d(x^4+1)/d(x)

4. Differentiate the innermost polynomial x^4+1 using the power rule.

d(x^4+1)/d(x)=4*x^3

5. Multiply all the components together to find the final derivative.

2*tan(x^4+1)*sec^2(x^4+1)⋅4*x^3

6. Simplify the expression by multiplying the constants and the power of x

8*x^3*tan(x^4+1)*sec^2(x^4+1)

Final Answer

d(tan(x^4+1))/d(x)=8*x^3*tan(x^4+1)*sec^2(x^4+1)

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