Calculus Mastery: Derivatives of Trigonometric and Inverse Trigonometric Functions

1) Differentiate sin x with respect to x.

Sin x

Tan x

Cos x

X

The derivative of sin x with respect to x is cos x due to the derivative of sin x being cos x. The incorrect answers do not represent the correct derivative of sin x.

Explanation

The derivative of sin x with respect to x is cos x due to the derivative of sin x being cos x. The incorrect answers do not represent the correct derivative of sin x.

2) What is the derivative of cos x with respect to x?

Tan x

-sin x

-cos x

Cos x

The derivative of cos x with respect to x is -sin x. This can be obtained using the chain rule of differentiation applied to the basic trigonometric derivative formula.

Explanation

The derivative of cos x with respect to x is -sin x. This can be obtained using the chain rule of differentiation applied to the basic trigonometric derivative formula.

3) Calculate the derivative of tan(x).

Cos^2(x)

Cot(x)

Sin^2(x)

Sec2x

The derivative of tan(x) is sec^2(x) because the derivative of tan(x) is equal to sec^2(x). The incorrect answers provided are not the correct derivative of tan(x).

Explanation

The derivative of tan(x) is sec^2(x) because the derivative of tan(x) is equal to sec^2(x). The incorrect answers provided are not the correct derivative of tan(x).

4) Differentiate cot x with respect to x.

Sec^2x

-cscx

Tan^2x

-csc2x

The derivative of cot x is -csc^2x, obtained by using the chain rule and the fact that cot x = 1/tan x = cos x/sin x. This results in the derivative of -sin x/cos x = -(-sin^2x / cos2x) = -csc^2x.

Explanation

The derivative of cot x is -csc^2x, obtained by using the chain rule and the fact that cot x = 1/tan x = cos x/sin x. This results in the derivative of -sin x/cos x = -(-sin^2x / cos2x) = -csc^2x.

5) Find the derivative of sec(x).

Cos(x) sin(x)

Sec x tan x

1/cos(x)

Cot(x) csc(x)

The derivative of sec(x) is sec(x) tan(x) because it is derived from the differentiation rules of trigonometric functions.

Explanation

The derivative of sec(x) is sec(x) tan(x) because it is derived from the differentiation rules of trigonometric functions.

6) What is the derivative of csc x?

Sin x

Csc x cot x

-csc x cot x

Sec x tan x

The derivative of csc x is -csc x cot x, which can be obtained using the chain rule for differentiation.

Explanation

The derivative of csc x is -csc x cot x, which can be obtained using the chain rule for differentiation.

7) Find the derivative of cos^{-1}u with respect to x.

-\frac{1}{\sqrt{1-u^2}}

-\frac{1}{\sqrt{1+u^2}}

-1/(sqrt(1-u2) * du/dx

-\frac{u}{\sqrt{1-u^2}}

The derivative of cos^{-1}u with respect to x is given by -\frac{1}{\sqrt{1-u^2}}\frac{du}{dx}. This is derived using the chain rule and the derivative of cos^{-1}u with respect to u.

Explanation

The derivative of cos^{-1}u with respect to x is given by -\frac{1}{\sqrt{1-u^2}}\frac{du}{dx}. This is derived using the chain rule and the derivative of cos^{-1}u with respect to u.

8) Find d/dx (cot-1u).

1/(1-u^2) * du/dx

-(1+u^2) * du/dx

-1/(1+u2) * du/dx

-1/(1-u^2) * du/dx

1/(1+u^2) * du/dx

The correct differentiation of the inverse cotangent function is -1/(1+u^2) * du/dx. This is derived using the chain rule and the derivative of the inverse cotangent function. The incorrect answers provide variations that do not align with the differentiation rules for inverse trigonometric functions.

Explanation

The correct differentiation of the inverse cotangent function is -1/(1+u^2) * du/dx. This is derived using the chain rule and the derivative of the inverse cotangent function. The incorrect answers provide variations that do not align with the differentiation rules for inverse trigonometric functions.

9) Find the derivative of sin-1u with respect to x.

1/(sqrt(1-u2) * du/dx

D/sin(u)

1/(u^2) * du/dx

Cos-1u

The correct derivative of sin-1u with respect to x is calculated using the chain rule to be 1/(sqrt(1-u^2)) times the derivative of u with respect to x, denoted as du/dx.

Explanation

The correct derivative of sin-1u with respect to x is calculated using the chain rule to be 1/(sqrt(1-u^2)) times the derivative of u with respect to x, denoted as du/dx.

10) Find the derivative of tan-1 u with respect to x.

Tan(u) * du/dx

1/(1+u2) * du/dx

1/u^2 * du/dx

Cos(u) * du/dx

The correct derivative of tan-1 u can be found using the formula 1/(1+u^2) * du/dx, which is a result of the derivative of the inverse of tan function. The other provided incorrect answers do not correctly represent the derivative of tan-1 u.

Explanation

The correct derivative of tan-1 u can be found using the formula 1/(1+u^2) * du/dx, which is a result of the derivative of the inverse of tan function. The other provided incorrect answers do not correctly represent the derivative of tan-1 u.

11) Find the derivative of sec^-1(u) with respect to x.

1 / (u * sqrt(u^2 - 1))

1/(abs(u) * sqrt(u2-1) * du/dx

1 / (u * sqrt(u^2 + 1) * du/dx)

1 / (u * sqrt(u^2 - 1) * u^2)

To find the derivative of sec^-1(u) with respect to x, we use the chain rule of differentiation and plug in the derivative of sec^-1(u) which is 1 / (|u| * sqrt(u^2 - 1)) multiplied by du/dx.

Explanation

To find the derivative of sec^-1(u) with respect to x, we use the chain rule of differentiation and plug in the derivative of sec^-1(u) which is 1 / (|u| * sqrt(u^2 - 1)) multiplied by du/dx.

12) Find the derivative of csc^-1(u) with respect to x.

-1/(abs(u)*sqrt(u2-1)) * du/dx

1/(u*sqrt(u^2-1)) * du/dx

-1/(|u|*sqrt(u^2+1)) * du/dx

1/(u*sqrt(1-u^2)) * du/dx

The correct answer comes from the derivative of the inverse cosecant function, csc^-1(u), which is -1/(|u|*sqrt(u^2-1)). The chain rule is applied to find the final expression for the derivative with respect to x.

Explanation

The correct answer comes from the derivative of the inverse cosecant function, csc^-1(u), which is -1/(|u|*sqrt(u^2-1)). The chain rule is applied to find the final expression for the derivative with respect to x.