Find the Derivative - d/dx y=16x^(1/2)+4x

Problem

d()/d(x)*(16*x(1/2)+4*x)

Solution

1. Apply the sum rule for differentiation, which allows for taking the derivative of each term individually.

d()/d(x)*(16*x(1/2)+4*x)=(d(16)*x(1/2))/d(x)+(d(4)*x)/d(x)

2. Apply the power rule to the first term, where (d(a)*xn)/d(x)=a⋅n⋅x(n−1)

(d(16)*x(1/2))/d(x)=16⋅1/2⋅x(1/2−1)

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

16⋅1/2⋅x(−1/2)=8*x(−1/2)

4. Apply the power rule to the second term.

(d(4)*x)/d(x)=4

5. Combine the results of the derivatives.

8*x(−1/2)+4

6. Rewrite the expression using radical notation if desired, noting that x(−1/2)=1/√(,x)

8/√(,x)+4

Final Answer

d()/d(x)*(16*x(1/2)+4*x)=8/√(,x)+4

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