Find the Derivative - d/dx 6 square root of x+5cos(x)

Problem

d()/d(x)*(6√(,x)+5*cos(x))

Solution

1. Apply the sum rule for derivatives to differentiate each term separately.

d()/d(x)*(6√(,x)+5*cos(x))=(d(6)√(,x))/d(x)+(d(5)*cos(x))/d(x)

2. Rewrite the radical using a fractional exponent to prepare for the power rule.

√(,x)=x(1/2)

3. Apply the constant multiple rule to move the coefficients outside the derivatives.

6d(x(1/2))/d(x)+5d(cos(x))/d(x)

4. Differentiate the power function using the power rule d(xn)/d(x)=n*x(n−1)

d(x(1/2))/d(x)=1/2*x(−1/2)

5. Differentiate the trigonometric function using the rule d(cos(x))/d(x)=−sin(x)

d(cos(x))/d(x)=−sin(x)

6. Combine and simplify the results by multiplying the constants and rewriting the negative exponent.

6*(1/2*x(−1/2))+5*(−sin(x))=3*x(−1/2)−5*sin(x)

7. Convert back to radical form for the final expression.

3*x(−1/2)=3/√(,x)

Final Answer

d()/d(x)*(6√(,x)+5*cos(x))=3/√(,x)−5*sin(x)

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