Find the Derivative - d/dx

Problem

d()/d(x)*(sin(√(3,x))+√(3,sin(5*x)))

Solution

1. Rewrite the expression using fractional exponents to make differentiation easier.

sin(x(1/3))+(sin(5*x))(1/3)

2. Apply the sum rule to differentiate each term separately.

d(sin(x(1/3)))/d(x)+d(sin(5*x))/d(x)

3. Apply the chain rule to the first term, sin(x(1/3))

cos(x(1/3))⋅d(x(1/3))/d(x)

cos(x(1/3))⋅1/3*x(−2/3)

4. Apply the chain rule to the second term, (sin(5*x))(1/3) which requires multiple steps.

1/3*(sin(5*x))(−2/3)⋅d(sin(5*x))/d(x)

1/3*(sin(5*x))(−2/3)⋅cos(5*x)⋅(d(5)*x)/d(x)

1/3*(sin(5*x))(−2/3)⋅cos(5*x)⋅5

5. Combine and simplify the results from both terms.

cos(x(1/3))/(3*x(2/3))+(5*cos(5*x))/(3*(sin(5*x))(2/3))

Final Answer

d()/d(x)*(sin(√(3,x))+√(3,sin(5*x)))=cos(√(3,x))/(3√(3,x2))+(5*cos(5*x))/(3√(3,sin2(5*x)))

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