Find dy/dx y=16 fourth root of 4x^4+4

Problem

y=16√(4,4*x4+4)

Solution

1. Rewrite the expression using a fractional exponent to prepare for differentiation.

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

2. Apply the chain rule by differentiating the outer function while keeping the inner function the same, then multiplying by the derivative of the inner function.

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

3. Simplify the coefficient and the exponent of the first part.

d(y)/d(x)=4*(4*x4+4)(−3/4)⋅d(4*x4+4)/d(x)

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

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

5. Combine the results and simplify the expression.

d(y)/d(x)=4*(4*x4+4)(−3/4)⋅16*x3

6. Multiply the constants and move the negative exponent to the denominator.

d(y)/d(x)=(64*x3)/((4*x4+4)(3/4))

Final Answer

d(y)/d(x)=(64*x3)/((4*x4+4)(3/4))

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