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

Problem

d()/d(x)*16√(4,4*x4+4)

Solution

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

16√(4,4*x4+4)=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()/d(x)*16*(4*x4+4)(1/4)=16⋅1/4*(4*x4+4)(−3/4)⋅d()/d(x)*(4*x4+4)

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

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

4. Substitute the derivative of the inner function back into the expression.

4*(4*x4+4)(−3/4)⋅16*x3

5. Simplify the expression by multiplying the constants and moving the negative exponent to the denominator.

(64*x3)/((4*x4+4)(3/4))

6. Factor the constant 4 out of the expression inside the denominator to simplify further.

(4*(x4+1))(3/4)=4(3/4)*(x4+1)(3/4)

7. Simplify the constants using the fact that 4(3/4)=(2)(3/4)=2(3/2)=2√(,2)

(64*x3)/(2√(,2)*(x4+1)(3/4))

8. Rationalize or simplify the numerical coefficient.

(32*x3)/(√(,2)*(x4+1)(3/4))=(16√(,2)*x3)/((x4+1)(3/4))

Final Answer

d()/d(x)*16√(4,4*x4+4)=(16√(,2)*x3)/((x4+1)(3/4))

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