The derivative formulas for the six basic trigonometric functions are:
To use the chain rule to find derivatives of trigonometric functions, you must first rewrite the function in terms of a single variable. Then, take the derivative of the outer function and multiply it by the derivative of the inner function. For example, if you have the function f(x) = sin(3x), you would rewrite it as f(u) = sin(u) where u = 3x. Then, using the derivative formula for sin x, you would get f'(u) = cos(u) = cos(3x). Finally, multiply by the derivative of the inner function (3) to get the final answer: f'(x) = 3cos(3x).
The relationship between the derivatives of sine and cosine is that the derivative of cosine is equal to the negative derivative of sine. In other words, the derivative of cosine is the negative of the derivative of sine. This relationship can be seen in the derivative formulas for sine and cosine: cos x = -sin x and sin x = cos x.
To find the derivative of a trigonometric function using the quotient rule, you must first rewrite the function as a fraction. Then, apply the quotient rule, which states that the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator, all divided by the denominator squared. For example, if you have the function f(x) = tan x / cos x, you would rewrite it as f(x) = tan x * cos^-1 x. Then, using the quotient rule, you would get f'(x) = (sec^2 x * cos x - tan x * -sin x) / cos^2 x = sec^2 x + tan x * sin x / cos^2 x.
The derivative of the inverse trigonometric functions is given by the formula: f'(x) = 1 / √(1 - x^2). This applies to the inverse sine, inverse cosine, and inverse tangent functions. The derivative of the inverse cotangent function is -1 / √(1 - x^2), and the derivative of the inverse secant and inverse cosecant functions is 1 / (|x| * √(x^2 - 1)).