Find the Derivative - d/d@VAR f(x)=x^5cos(x)

Problem

d()/d(x)*x5*cos(x)

Solution

1. Identify the function as a product of two terms, u=x5 and v=cos(x) which requires the product rule.

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

3. Differentiate the individual components: d(x5)/d(x)=5*x4 and d(cos(x))/d(x)=−sin(x)

4. Substitute these derivatives back into the product rule formula to get x5*(−sin(x))+cos(x)*(5*x4)

5. Simplify the expression by rearranging the terms and factoring out common factors if necessary.

Final Answer

d()/d(x)*x5*cos(x)=5*x4*cos(x)−x5*sin(x)

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