Find the Derivative - d/dx y=x^2sin(x)^4+cos(x)^-2

Problem

d()/d(x)*(x^2*sin^4(x)+cos(x)^(−2))

Solution

1. Identify the structure of the expression as a sum of two terms, x^2*sin^4(x) and cos(x)^(−2) and apply the sum rule for differentiation.

2. Apply the product rule to the first term x^2*sin^4(x) which states (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

3. Apply the chain rule to differentiate sin^4(x) resulting in 4*sin^3(x)*cos(x)

4. Combine the results for the first term to get x^2*(4*sin^3(x)*cos(x))+2*x*sin^4(x)

5. Apply the chain rule to the second term cos(x)^(−2) treating it as u^(−2) where u=cos(x)

6. Differentiate the second term to get −2*cos(x)^(−3)*(−sin(x)) which simplifies to 2*sin(x)*cos(x)^(−3)

7. Simplify the final expression by combining all differentiated parts.

Final Answer

d()/d(x)*(x^2*sin^4(x)+cos(x)^(−2))=4*x^2*sin^3(x)*cos(x)+2*x*sin^4(x)+2*sin(x)*sec^3(x)

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