Find the Derivative - d/dx y=16 fourth root of 4x^4+4

Problem

d()/d(x)*16√(4,4*x4+4)

Solution

1. Rewrite the expression using a fractional exponent to make it easier to differentiate.

y=16*(4*x4+4)1/4

2. Apply the chain rule, which states that the derivative of ƒ*(g(x)) is ƒ′*(g(x))⋅g(x)′

d(y)/d(x)=16⋅1/4*(4*x4+4)(1/4−1)⋅d()/d(x)*(4*x4+4)

3. Differentiate the inner function 4*x4+4 using the power rule.

d(4*x4+4)/d(x)=16*x3

4. Substitute the inner derivative back into the expression and simplify the coefficients.

d(y)/d(x)=4*(4*x4+4)(−3/4)⋅16*x3

5. Multiply the remaining terms to find the final derivative.

d(y)/d(x)=64*x3*(4*x4+4)(−3/4)

6. Rewrite the expression using radical notation and positive exponents.

d(y)/d(x)=(64*x3)/√(4,(4*x4+4)3)

Final Answer

d()/d(x)*16√(4,4*x4+4)=(64*x3)/√(4,(4*x4+4)3)

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