Find the Derivative - d/dx x^9cos(x)

Problem

d()/d(x)*x9*cos(x)

Solution

1. Identify the rule needed for the derivative. Since the expression is a product of two functions, x9 and cos(x) use the product rule: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Assign the functions to the variables u and v Let u=x9 and v=cos(x)

3. Differentiate each part individually. The derivative of x9 is 9*x8 using the power rule, and the derivative of cos(x) is −sin(x)

4. Apply the product rule by substituting these components into the formula.

(d(x9)*cos(x))/d(x)=x9d(cos(x))/d(x)+cos(x)d(x9)/d(x)

5. Substitute the derivatives found in step 3.

(d(x9)*cos(x))/d(x)=x9*(−sin(x))+cos(x)*(9*x8)

6. Simplify the expression by rearranging the terms and factoring out the greatest common factor, x8

(d(x9)*cos(x))/d(x)=−x9*sin(x)+9*x8*cos(x)

(d(x9)*cos(x))/d(x)=x8*(9*cos(x)−x*sin(x))

Final Answer

(d(x9)*cos(x))/d(x)=x8*(9*cos(x)−x*sin(x))

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