Find the Derivative - d/dx y=5xe^x-5e^x

Problem

d()/d(x)*(5*x*ex−5*ex)

Solution

1. Apply the sum rule to differentiate each term of the expression separately.

d()/d(x)*(5*x*ex−5*ex)=(d(5)*x*ex)/d(x)−(d(5)*ex)/d(x)

2. Apply the product rule to the first term 5*x*ex where the derivative of u*v is ud(v)/d(x)+vd(u)/d(x)

(d(5)*x*ex)/d(x)=5*xd(ex)/d(x)+ex(d(5)*x)/d(x)

3. Differentiate the components of the product rule expression.

(d(5)*x*ex)/d(x)=5*x(ex)+e(5)x

4. Differentiate the second term of the original expression using the constant multiple rule.

(d(5)*ex)/d(x)=5*ex

5. Combine the results from the previous steps.

d()/d(x)*(5*x*ex−5*ex)=5*x*ex+5*ex−5*ex

6. Simplify the expression by canceling the opposite terms.

d()/d(x)*(5*x*ex−5*ex)=5*x*ex

Final Answer

d()/d(x)*(5*x*ex−5*ex)=5*x*ex

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