Trigonometric Derivatives Quiz

The derivative of sin(x) is cos(x).

The derivative of the cosine function, y = cos(x), is equal to the negative sine function, y' = -sin(x). This can be derived using the chain rule of differentiation.

The derivative of y = tan(x) is y' = sec^2(x). This is because the derivative of tan(x) is equal to the derivative of sin(x)/cos(x), which can be rewritten as 1/cos^2(x). Since 1/cos^2(x) is equivalent to sec^2(x), the correct answer is y' = sec^2(x).

The derivative of y = sec(x) is y' = sec(x)tan(x). This can be derived using the quotient rule, where the derivative of sec(x) is sec(x)tan(x) and the derivative of tan(x) is sec^2(x). Therefore, the correct answer is y' = sec(x)tan(x).

The derivative of y=cot(x) is y'=-csc^2(x). This is because the derivative of cot(x) can be found using the quotient rule, where the derivative of the numerator is -csc^2(x) and the derivative of the denominator is 1. Therefore, the derivative of cot(x) is -csc^2(x).

The correct answer is y'=-Csc(x)Cot(x). The derivative of y = csc(x) can be found using the chain rule. The derivative of csc(x) is -csc(x)cot(x), where -csc(x) represents the derivative of the outer function and cot(x) represents the derivative of the inner function. Therefore, the correct answer is y'=-Csc(x)Cot(x).

The given function is y = sin^(-1)(x), which represents the inverse sine function. To find the derivative of this function, we can use the chain rule. The derivative of sin^(-1)(x) is 1/√(1-x^2). Therefore, the correct answer is y' = 1/√(1-x^2).

The given function is y = Tan^-1(x), which represents the inverse tangent of x. The derivative of the inverse tangent function is 1/(1+x^2). This can be derived using the chain rule and the derivative of the tangent function. Therefore, the correct answer is y' = 1/(1+x^2).

The given correct answer is y' = -1/√(1-x^2). This can be derived using the chain rule of differentiation. The derivative of y = Cos^-1(x) with respect to x can be found by differentiating the inverse cosine function. Applying the chain rule, we have y' = -1/√(1-x^2), which matches the given correct answer.

The given correct answer is y' = -1/(1+x^2). This can be derived using the chain rule of differentiation. The derivative of cot^-1(x) can be found by differentiating the inverse cotangent function. The derivative of cot^-1(x) is equal to -1/(1+x^2). Therefore, the correct answer is y' = -1/(1+x^2).

The given answer, y' = -1/(x√(1-x^2), is the derivative of the function y = csc^-1(x). This can be determined using the chain rule and the derivative of the inverse cosecant function. The derivative of csc^-1(x) is equal to -1/(x√(1-x^2), which matches the given answer.

The given expression is y = Sec^-1(x), which represents the inverse secant function. To find the derivative of this function, we can use the chain rule. The derivative of Sec^-1(x) is equal to 1 divided by the square root of (1 - x^2). Therefore, the correct answer is y' = 1/(x√(1-x^2)).