Find the Derivative - d/dx 7 square root of xsin(x)

Problem

d()/d(x)*7√(,x)*sin(x)

Solution

1. Identify the constant and the product of functions. The constant 7 can be moved outside the derivative, and the remaining expression √(,x)*sin(x) requires the product rule.

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

√(,x)=x(1/2)

3. Apply the product rule, which states d()/d(x)*ƒ(x)*g(x)=ƒ(x)′*g(x)+ƒ(x)*g(x)′ Let ƒ(x)=x(1/2) and g(x)=sin(x)

(d(7)*x(1/2)*sin(x))/d(x)=7*(d(x(1/2))/d(x)*sin(x)+x(1/2)d(sin(x))/d(x))

4. Differentiate the individual components using the power rule for x(1/2) and the trigonometric derivative for sin(x)

d(x(1/2))/d(x)=1/2*x(−1/2)

d(sin(x))/d(x)=cos(x)

5. Substitute these derivatives back into the product rule expression.

7*(1/2*x(−1/2)*sin(x)+x(1/2)*cos(x))

6. Simplify the expression by distributing the 7 and converting the fractional powers back into radical form.

(7*sin(x))/(2√(,x))+7√(,x)*cos(x)

Final Answer

(d(7)√(,x)*sin(x))/d(x)=(7*sin(x))/(2√(,x))+7√(,x)*cos(x)

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