Derivative of \( \displaystyle \operatorname{atan}{\left(\left(5 x - 3\right)^{2} \right)} \)

Problem 2.326 · hard

1. \[ \frac{d}{d x} \operatorname{atan}{\left(\left(5 x - 3\right)^{2} \right)} \]
   rewriteStart with the derivative of the function. Rewrite the power using the exponential and logarithm identity.✓ Proved
2. \[ = \frac{\frac{d}{d x} \left(5 x - 3\right)^{2}}{\left(5 x - 3\right)^{4} + 1} \]
   chain algebraApply the chain rule for the arctangent function. Simplify the denominator by converting the exponential back to a power.✓ Proved
3. \[ = \frac{2 \left(5 x - 3\right)^{2} \frac{d}{d x} \ln{\left(5 x - 3 \right)}}{\left(5 x - 3\right)^{4} + 1} \]
   chainApply the chain rule to the numerator.✓ Proved
4. \[ = \frac{50 x - 30}{\left(5 x - 3\right)^{4} + 1} \]
   chain constant-multiple algebra simplifyDifferentiate the logarithm term using the chain rule. Simplify the constants in the numerator. Convert the exponential back to a power for simplification. Simplify the fraction by canceling the common term (5*x - 3).✓ Proved