Find the Derivative - d/dx y=9xe^(-kx)

Problem

d()/d(x)*9*x*e(−k*x)

Solution

1. Identify the rule needed for differentiation. Since the expression 9*x*e(−k*x) is a product of two functions, 9*x and e(−k*x) the product rule d()/d(x)*[u*v]=ud(v)/d(x)+vd(u)/d(x) must be used.

2. Assign the parts of the product rule where u=9*x and v=e(−k*x)

3. Differentiate each part individually. The derivative of u is (d(9)*x)/d(x)=9 To differentiate v apply 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 back in.

d(y)/d(x)=(9*x)*(−k*e(−k*x))+(e(−k*x))*(9)

5. Simplify the expression by factoring out the common terms 9 and e(−k*x)

d(y)/d(x)=−9*k*x*e(−k*x)+9*e(−k*x)

d(y)/d(x)=9*e(−k*x)*(1−k*x)

Final Answer

d()/d(x)*9*x*e(−k*x)=9*e(−k*x)*(1−k*x)

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