Derivative of \( \displaystyle x \sin^{2}{\left(4 x - 1 \right)} + x \cos^{2}{\left(4 x - 1 \right)} - \frac{\sin{\left(8 x - 2 \right)}}{8} \)

Problem 2.1072 · hard

1. \[ \frac{d}{d x} \left(x \sin^{2}{\left(4 x - 1 \right)} + x \cos^{2}{\left(4 x - 1 \right)} - \frac{\sin{\left(8 x - 2 \right)}}{8}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} x \sin^{2}{\left(4 x - 1 \right)} + \frac{d}{d x} x \cos^{2}{\left(4 x - 1 \right)} - \frac{d}{d x} \frac{\sin{\left(8 x - 2 \right)}}{8} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} x \sin^{2}{\left(4 x - 1 \right)} + \frac{d}{d x} x \cos^{2}{\left(4 x - 1 \right)} - \frac{\frac{d}{d x} \sin{\left(8 x - 2 \right)}}{8} \]
   constant-multipleFactor out the constant 1/8.✓ Proved
4. \[ = - \frac{\cos{\left(8 x - 2 \right)} \frac{d}{d x} \left(8 x - 2\right)}{8} + \frac{d}{d x} x \sin^{2}{\left(4 x - 1 \right)} + \frac{d}{d x} x \cos^{2}{\left(4 x - 1 \right)} \]
   chainApply the chain rule to the sine term.✓ Proved
5. \[ = - \cos{\left(8 x - 2 \right)} + \frac{d}{d x} x \sin^{2}{\left(4 x - 1 \right)} + \frac{d}{d x} x \cos^{2}{\left(4 x - 1 \right)} \]
   derivative algebraDifferentiate the inner linear function. Simplify the constant multiplication.✓ Proved
6. \[ = x \frac{d}{d x} \sin^{2}{\left(4 x - 1 \right)} + x \frac{d}{d x} \cos^{2}{\left(4 x - 1 \right)} + \sin^{2}{\left(4 x - 1 \right)} \frac{d}{d x} x + \cos^{2}{\left(4 x - 1 \right)} \frac{d}{d x} x - \cos{\left(8 x - 2 \right)} \]
   productApply the product rule to both terms.✓ Proved
7. \[ = x \frac{d}{d x} \sin^{2}{\left(4 x - 1 \right)} + x \frac{d}{d x} \cos^{2}{\left(4 x - 1 \right)} + \sin^{2}{\left(4 x - 1 \right)} + \cos^{2}{\left(4 x - 1 \right)} - \cos{\left(8 x - 2 \right)} \]
   derivative algebraDifferentiate the x terms. Simplify the coefficients.✓ Proved
8. \[ = \sin^{2}{\left(4 x - 1 \right)} + \cos^{2}{\left(4 x - 1 \right)} - \cos{\left(8 x - 2 \right)} \]
   chain derivative algebra algebraApply the chain rule to the squared trigonometric terms. Differentiate the inner linear functions. Multiply the constants. Cancel the identical terms with opposite signs.✓ Proved
9. \[ = 1 - \cos{\left(8 x - 2 \right)} \]
   simplifyUse the trigonometric identity sin^2(u) + cos^2(u) = 1.✓ Proved