Find the Derivative - d/dx f(x)=x square root of x+4

Problem

d()/d(x)*x√(,x+4)

Solution

1. Rewrite the square root as a fractional power to make differentiation easier.

ƒ(x)=x*(x+4)(1/2)

2. Apply the product rule, which states that d()/d(x)*u*v=ud(v)/d(x)+vd(u)/d(x) where u=x and v=(x+4)(1/2)

d(ƒ(x))/d(x)=xd(x+4)/d(x)+(x+4)(1/2)d(x)/d(x)

3. Apply the chain rule to differentiate (x+4)(1/2)

d(x+4)/d(x)=1/2*(x+4)(−1/2)⋅1

4. Substitute the derivatives back into the product rule expression.

d(ƒ(x))/d(x)=x⋅1/2*(x+4)(−1/2)+(x+4)(1/2)⋅1

5. Simplify the expression by finding a common denominator to combine the terms.

d(ƒ(x))/d(x)=x/(2√(,x+4))+√(,x+4)

6. Combine the terms over the common denominator 2√(,x+4)

d(ƒ(x))/d(x)=(x+2*(x+4))/(2√(,x+4))

7. Distribute and combine like terms in the numerator.

d(ƒ(x))/d(x)=(x+2*x+8)/(2√(,x+4))

Final Answer

d()/d(x)*x√(,x+4)=(3*x+8)/(2√(,x+4))

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