Derivatives of Trigonometric Functions

• The derivative of sine x is equal to cosine x. • The derivative of cosine x is equal to negative sine x. • The derivative of tangent x is equal to positive secant squared. • The derivative of cosecant x is equal to negative cosecant x cotangent x. • The derivative of secant x is equal to positive secant x tan x. • The derivative of cotangent x is equal to negative cosecant squared x. • When dealing with trigonometric functions, every time there's a c (cosecant, cosine, or cotangent), there's usually a negative in front, which can help with memorization. • The power rule states that the derivative of x^n is n*x^(n-1), where n is a constant. • To find the derivative of a function, apply the power rule and the derivatives of trigonometric functions. • The product rule states that if f(x) = u(x) * v(x), then f'(x) = u'(x) * v(x) + u(x) * v'(x). • Using the product rule, the derivative of x^2 * sine x is 2x * sine x + x^2 * cosine x. • The quotient rule states that if f(x) = u(x) / v(x), then f'(x) = (v(x) * u'(x) - u(x) * v'(x)) / v(x)^2. • Using the quotient rule, the derivative of sine x / x^2 is (x^2 * cosine x - 2 * sine x) / x^4. • The chain rule states that if f(x) = f(g(x)), then f'(x) = f'(g(x)) * g'(x). • The chain rule can be applied to composite functions, such as sine(u) or cosine(u), where u is a function of x. • The derivative of sine(u) is cosine(u) * u', and the derivative of cosine(u) is -sine(u) * u'. • The derivatives of other trigonometric functions, such as tangent(u), cosecant(u), secant(u), and cotangent(u), can also be found using the chain rule. • The derivative of sine(5x) is cosine(5x) * 5, using the chain rule and the derivative of sine(u).- To find the derivative of a composite function, differentiate the outside function first, then multiply it by the derivative of the inside function.

• The derivative of cosine is negative sine.
• The derivative of secant is secant tangent.
• The derivative of cotangent is negative cosecant squared.
• The derivative of sine is cosine.
• The derivative of tangent is secant squared.
• The derivative of f(x) = cosine(x)³ is -3x² sine(x)³.
• The derivative of f(x) = secant(x²) is 2x secant(x²) tan(x²).
• The derivative of f(x) = tangent(sine(4x)) is 4 cosine(4x) secant²(sine(4x)).
• The derivative of f(x) = sine²(3x) is 6 sine(3x) cosine(3x).
• The derivative of f(x) = cotangent(sine(x³))⁴ is -12x² cosine(x³) cosecant²(sine(x³)) cotangent(sine(x³))³.
• The derivative of f(x) = sine(x³)² is 2 sine(x³) cosine(x³)³.
• The derivative of f(x) = cosine(tangent(sine(x)))⁵ is -25x⁴ cosine(x) secant²(tangent(sine(x)))⁴.
• To prove that the derivative of secant(x) is secant(x) tangent(x), use the power rule and the chain rule on secant(x) = 1/cosine(x).
• To prove that the derivative of cotangent(x) is -cosecant²(x), use the quotient rule on cotangent(x) = cosine(x)/sine(x).