Find the Derivative - d/dz (4z+e^(-z^2))^5

Problem

d()/d(z)*(4*z+e^(−z^2))^5

Solution

1. Identify the outer function and the inner function to apply the Chain Rule, which states d()/d(z)*[ƒ*(g(z))]=ƒ^′*(g(z))⋅g(z)^′

2. Apply the Power Rule to the outer expression by bringing the exponent 5 to the front and decreasing the power by 1

5*(4*z+e^(−z^2))^4⋅d()/d(z)*(4*z+e^(−z^2))

3. Differentiate the inner expression term by term using the Sum Rule.

d()/d(z)*(4*z)+d()/d(z)*e^(−z^2)

4. Apply the Chain Rule again to the exponential term e^(−z^2) where the derivative of e^u is e^u⋅d(u)/d(z)

d(e^(−z^2))/d(z)=e^(−z^2)⋅(d(−)*z^2)/d(z)

5. Calculate the derivatives of the individual components: (d(4)*z)/d(z)=4 and (d(−)*z^2)/d(z)=−2*z

4+e^(−z^2)*(−2*z)

6. Combine all parts back into the final derivative expression.

5*(4*z+e^(−z^2))^4*(4−2*z*e^(−z^2))

Final Answer

d()/d(z)*(4*z+e^(−z^2))^5=5*(4*z+e^(−z^2))^4*(4−2*z*e^(−z^2))

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