Derivatives of Trigonometric Functions

Study Notes

• 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).

Description

Test your understanding of derivatives of trigonometric functions such as sine, cosine, and tangent, as well as the power rule, product rule, quotient rule, and chain rule. Learn how to apply these rules to find derivatives of composite functions.