Find the Derivative - d/dx y=4xe^(-kx)

Problem

d()/d(x)*4*x*e(−k*x)

Solution

1. Identify the rule needed for differentiation. Since the expression is a product of two functions, 4*x and e(−k*x) the product rule (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x) must be used.

2. Assign the parts of the product. Let u=4*x and v=e(−k*x)

3. Differentiate each part individually. The derivative of u is (d(4)*x)/d(x)=4 The derivative of v requires the chain rule: d(e(−k*x))/d(x)=e(−k*x)⋅(d(−)*k*x)/d(x)=−k*e(−k*x)

4. Apply the product rule formula by substituting the parts.

d(y)/d(x)=4*x*(−k*e(−k*x))+e(4)(−k*x)

5. Simplify the expression by factoring out common terms. Both terms contain 4 and e(−k*x)

d(y)/d(x)=4*e(−k*x)*(−k*x+1)

Final Answer

(d(4)*x*e(−k*x))/d(x)=4*e(−k*x)*(1−k*x)

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