Find the Derivative - d/dx e^(-(5-3x)^2)

Problem

d(e^(−(5−3*x)^2))/d(x)

Solution

1. Identify the outer function and the inner function to apply the Chain Rule. The outer function is e^u where u=−(5−3*x)^2

2. Differentiate the outer function e^u with respect to u which results in e^u

3. Apply the Chain Rule by multiplying the derivative of the outer function by the derivative of the exponent u=−(5−3*x)^2

d(e^(−(5−3*x)^2))/d(x)=e^(−(5−3*x)^2)⋅(d(−)*(5−3*x)^2)/d(x)

4. Differentiate the exponent using the Power Rule and the Chain Rule again.

(d(−)*(5−3*x)^2)/d(x)=−2*(5−3*x)⋅d(5−3*x)/d(x)

5. Calculate the derivative of the innermost linear expression (5−3*x) which is −3

d(5−3*x)/d(x)=−3

6. Combine all the components together.

d(e^(−(5−3*x)^2))/d(x)=e^(−(5−3*x)^2)⋅(−2*(5−3*x))⋅(−3)

7. Simplify the constant coefficients and the linear term.

−2⋅−3=6

6⋅(5−3*x)=30−18*x

Final Answer

d(e^(−(5−3*x)^2))/d(x)=(30−18*x)*e^(−(5−3*x)^2)

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