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

Problem

d()/d(x)*(16^4√(,4*x^4+4))

Solution

1. Identify the constant and the function to be differentiated. The term 16^4 is a constant coefficient and can be moved outside the derivative.

2. Rewrite the square root using a fractional exponent to make it easier to apply power rules.

y=16^4*(4*x^4+4)^(1/2)

3. Apply the chain rule by differentiating the outer function (the power of 1/2 and multiplying by the derivative of the inner function (4*x^4+4.

d(y)/d(x)=16^4⋅1/2*(4*x^4+4)^(−1/2)⋅d(4*x^4+4)/d(x)

4. Differentiate the inner expression 4*x^4+4 using the power rule.

d(4*x^4+4)/d(x)=16*x^3

5. Substitute the inner derivative back into the expression.

d(y)/d(x)=16^4⋅1/2*(4*x^4+4)^(−1/2)⋅16*x^3

6. Simplify the constants. Note that 16^4⋅16=16^5 and 16^5⋅1/2=1048576/2=524288

d(y)/d(x)=(524288*x^3)/((4*x^4+4)^(1/2))

7. Factor the denominator to simplify further. Since 4*x^4+4=4*(x^4+1) the square root is √(,4)√(,x^4+1)=2√(,x^4+1)

d(y)/d(x)=(524288*x^3)/(2√(,x^4+1))

8. Divide the numerator by 2.

d(y)/d(x)=(262144*x^3)/√(,x^4+1)

Final Answer

d()/d(x)*(16^4√(,4*x^4+4))=(262144*x^3)/√(,x^4+1)

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