Find the Derivative - d/dx x^5cos(x)

Problem

d()/d(x)*x^5*cos(x)

Solution

1. Identify the rule needed for the derivative of a product of two functions, u(x)=x^5 and v(x)=cos(x)

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

3. Differentiate the first part by finding d(x^5)/d(x)=5*x^4 using the power rule.

4. Differentiate the second part by finding d(cos(x))/d(x)=−sin(x)

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

x^5*(−sin(x))+cos(x)*(5*x^4)

6. Simplify the expression by rearranging the terms and factoring out the greatest common factor, x^4

5*x^4*cos(x)−x^5*sin(x)

Final Answer

d()/d(x)*x^5*cos(x)=5*x^4*cos(x)−x^5*sin(x)

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