Trigonometric derivatives - Solved Examples

Set A

Differentiate the following with the help of the trigonometric derivative rules, and all derivation rules studied before-with.

Question 1

x^2 + 2sin(x) - cos(x)

Solution:

Take the derivative of the expression,

d/(dx) (x^2 + 2sin(x) - cos(x))

Take the derivative of the sum term by term, in accordance with the derivative properties,

d/(dx) x^2 + d/(dx) 2sin(x) - d/(dx) cos(x)

By power rule and trigonometric derivative rules, the derivatives are:

2x + 2cos(x) - (-sin(x))

2x + 2cos(x) + sin(x)

Question 2

(x + cos(x))/(x - tan(x))

Solution: 

Apply the quotient rule to take the derivative of this expression,

(((x - tan(x)) * d/(dx) (x + cos(x))) - ((x + cos(x)) * d/(dx) (x - tan(x))))/((x - tan(x))^2)

Take the derivative of each term term by term, and by applying power rule and product rule, we get

d/(dx) (x + cos(x)) = 1 - sin(x)

d/(dx)  (x - tan(x)) = 1 - sec^2(x)

Substituting the derivatives in the original expression,

(((x - tan(x)) * (1 - sin(x))) - ((x + cos(x)) * (1 - sec^2(x))))/((x - tan(x))^2)

Question 3

5csc(x) + 6sin(x) + 1

Solution:

Take the derivative of the sum term by term,

d/(dx) (5csc(x) + 6sin(x) + 1)

d/(dx) 5csc(x) + d/(dx) 6sin(x) + d/(dx) 1

Factor out the constants from each derivative,

5 d/(dx) csc(x) + 6 d/(dx) sin(x) + d/(dx) 1

Apply trigonometric derivative, rules and power rule,

5 (-csc(x)cot(x)) + 6 cos(x) + 0

Simplify the expression,

-5 csc(x)cot(x)) + 6 cos(x)

Question 4:

x^2sec(x) + sin(x)cos(x)

Solution: 

Take the derivative of the sum term by term,

d/(dx) x^2sec(x) + d/(dx) sin(x)cos(x)

Apply product rule to each product,

(x^2 * d/(dx) sec(x) + sec(x) * d/(dx) x^2) + (sin(x) * d/(dx) cos(x) + cos(x) * d/(dx) sin(x))

Apply the trigonometric derivative rules and power rule,

(x^2 * sec(x)tan(x) + sec(x) * 2x) + (sin(x) *-sin(x) + cos(x) * cos(x))

Simplify the expression,

(x^2 sec(x)tan(x) + 2x sec(x)) + (-sin^2(x) + cos^2(x))

Question 5

(cot(x))/(x^2 + 3x) + (tan(x))/(x^3 + 4x)

Solution:

Take the derivative of the sum term by term,

d/(dx) (cot(x))/(x^2 + 3x) + d/(dx) (tan(x))/(x^3 + 4x)

Apply quotient rule to each derivative,

(((x^2 + 3x) * d/(dx) cot(x)) - (cot(x) * d/(dx) (x^2 + 3x)))/((x^2 + 3x)^2) + (((x^3 + 4x) * d/(dx) (tan(x))) - ((tan(x)) * d/(dx) (x^3 + 4x)))/((x^3 + 4x)^2)

Apply derivation rules and formulas,

(((x^2 + 3x) *(-csc^2(x))) - (cot(x) * (2x + 3)))/((x^2 + 3x)^2) + (((x^3 + 4x) * sec^2(x)) - ((tan(x)) * (3x^2 + 4)))/((x^3 + 4x)^2)

Set B 

Differentiate the following by the help of trigonometric derivative rules:

Question 1

5 sinh(x) + 2coth(x)

Solution:

d/(dx)5 sinh(x) + 2coth(x)

Take the derivative of the sum term by term,

d/(dx)5 sinh(x) + d/(dx) 2coth(x)

Factor out the constants,

5 d/(dx) sinh(x) + 2 d/(dx) coth(x)

Applying the trigonometric derivative rules,

5 cosh(x) + 2 (-sinh(x))

5 cosh(x) - 2 sinh(x) 

Question 2

x^2 + x tanh(x)

Solution:

d/(dx) x^2 + x tanh(x)

Take the derivative of the sum term by term,

d/(dx) x^2 + d/(dx) x tanh(x)

Apply power rule on the first term and product rule on the second one,

2x + x d/(dx) tanh(x) + tanh(x) d/(dx) x

Apply trigonometric derivative rules,

2x + x sech^2(x) + tanh(x) * 1

 Simplifying,

2x + x sech^2(x) + tanh(x)

Question 3

4 sin^-1(x) + 7cos^-1(x)

Solution

d/(dx) 4 sin^-1(x) + 7cos^-1(x)

Take the derivative of the sum term by term,

d/(dx) 4 sin^-1(x) + d/(dx) 7cos^-1(x)

 Factor out the constants,

4 d/(dx) sin^-1(x) + 7 d/(dx) cos^-1(x)

Apply the formulas for trigonometric functions:

4 * 1/(1 - x^2) + 7 * 1/(√(1 - x^2))

Simplify,

4/(1 - x^2) + 7/(√(1 - x^2))

Question 4

tan^-1(x)sec^-1(x)

Solution:

Take the derivative of the product by the product rule,

tan^-1(x) d/(dx)sec^-1(x) + sec^-1(x) d/(dx) tan^-1(x)

Apply the derivative formulas for inverse trigonometric functions,