Find dy/dx y=5x^2e^(3x)

Problem

d()/d(x)*5*x2*e(3*x)

Solution

1. Identify the rule needed for differentiation. Since the expression is a product of two functions, 5*x2 and e(3*x) use the product rule: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Assign the functions to variables. Let u=5*x2 and v=e(3*x)

3. Differentiate each part individually.

(d(5)*x2)/d(x)=10*x

d(e(3*x))/d(x)=3*e(3*x)

4. Apply the product rule formula by substituting the functions and their derivatives.

d(y)/d(x)=(5*x2)*(3*e(3*x))+(e(3*x))*(10*x)

5. Simplify the expression by multiplying the terms.

d(y)/d(x)=15*x2*e(3*x)+10*x*e(3*x)

6. Factor out the common terms 5*x*e(3*x) to reach the final form.

d(y)/d(x)=5*x*e(3*x)*(3*x+2)

Final Answer

d(y)/d(x)=5*x*e(3*x)*(3*x+2)

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