Derivative of \( \displaystyle \frac{3 \ln{\left(\tan^{2}{\left(5 x - 3 \right)} + 1 \right)}}{10} \)

Problem 2.168 · hard

1. \[ \frac{d}{d x} \frac{3 \ln{\left(\tan^{2}{\left(5 x - 3 \right)} + 1 \right)}}{10} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{3 \frac{d}{d x} \ln{\left(\tan^{2}{\left(5 x - 3 \right)} + 1 \right)}}{10} \]
   constant-multiplePull out the constant factor.✓ Proved
3. \[ = \frac{3 \frac{d}{d x} \left(\tan^{2}{\left(5 x - 3 \right)} + 1\right)}{10 \left(\tan^{2}{\left(5 x - 3 \right)} + 1\right)} \]
   logarithmicApply the chain rule for the logarithm.✓ Proved
4. \[ = \frac{3 \left(\frac{d}{d x} 1 + \frac{d}{d x} \tan^{2}{\left(5 x - 3 \right)}\right)}{10 \left(\tan^{2}{\left(5 x - 3 \right)} + 1\right)} \]
   sumDifferentiate the sum inside the parentheses.✓ Proved
5. \[ = \frac{3 \frac{d}{d x} \tan^{2}{\left(5 x - 3 \right)}}{10 \left(\tan^{2}{\left(5 x - 3 \right)} + 1\right)} \]
   constantThe derivative of a constant is zero.✓ Proved
6. \[ = \frac{3 \tan{\left(5 x - 3 \right)} \frac{d}{d x} \tan{\left(5 x - 3 \right)}}{5 \left(\tan^{2}{\left(5 x - 3 \right)} + 1\right)} \]
   powerApply the power rule to the squared term.✓ Proved
7. \[ = \frac{3 \tan{\left(5 x - 3 \right)} \sec^{2}{\left(5 x - 3 \right)} \frac{d}{d x} \left(5 x - 3\right)}{5 \left(\tan^{2}{\left(5 x - 3 \right)} + 1\right)} \]
   trigApply the chain rule for the tangent function.≈ Checked numerically
8. \[ = \frac{3 \tan{\left(5 x - 3 \right)} \sec^{2}{\left(5 x - 3 \right)}}{\tan^{2}{\left(5 x - 3 \right)} + 1} \]
   derivative algebra simplifyDifferentiate the innermost linear expression. Multiply the constants. Simplify the expression by canceling 10.✓ Proved
9. \[ = 3 \tan{\left(5 x - 3 \right)} \]
   rewrite simplifyUse the identity sec(u)**2 = 1 + tan(u)**2. Cancel the common factor (tan(5*x - 3)**2 + 1).≈ Checked numerically