Differentiating Reciprocated Trig Functions

Example 1:

Differentiate each of the following with respect to and simplify your answers.

a)

Single Step: Identify the angle and apply the chain rule process to the tan function.

Here, we are differentiating the function, so the angle is what’s inside the function. The is a multiplier so we write that down first, then differentiate the angle.

The angle is , which differentiates to , the function differentiates to . Then, the angle inside stays the same.

Multiplier Angle stays the same



Differentiated angle

b)

Single Step: Identify the angle and apply the chain rule process to the sec and cot functions.

Here, we are differentiating the and functions.

For , the is a multiplier so we write that down first. The angle is , which differentiates to , the function differentiates to . Then, the angle inside stays the same.

For , the is a multiplier so we write that down first. The angle is , which differentiates to , the function differentiates to . Then, the angle inside stays the same.

Angle stays Differentiated Angle stays
Multiplier the same angle the same



Differentiated angle Multiplier

c)

Step 1: Identify the type of function.

Although this looks like a trigonometric function, where there are powers, that becomes the primary function. This is a power function, and we need to rewrite the function as
.

Step 2: Identify the angle and apply the chain rule process to a power function.

First, the is a multiplier so we write that down first. The angle of a power function is what is inside the bracket. Here, it is , which differentiates to . This is a power function, so we bring down the power, knock off the power, then the angle inside stays the same.

Angle stays the same



Multiplier Differentiated angle Differentiated power function

Example 2:

Differentiate each of the following with respect to and simplify your answers.

a)

Single Step: Identify the angle and apply the chain rule process to an exponential function.

Here, we are differentiating an exponential function, so the angle is the power. We first differentiate the angle, then the function of stays the same in differentiation as well as its power (in other words, the angle never changes).

Here, the angle is , so we differentiate that using the chain rule for the function. The is a multiplier so we write that at the front.

Multiplier Angle stays the same



Differentiated angle

The angle differentiates to give . So,

Angle stays the same



Differentiated angle

b)

Single Step: Identify the angle and apply the chain rule process to the cot function.

Here, we are differentiating the function, so the angle is inside the bracket. We first differentiate the angle, then the cot function differentiates to , then the angle inside doesn’t change.

Here, the angle is , so we differentiate that using the chain rule for the exponential function.

Angle stays the same



Differentiated angle

The angle differentiates to give . So,

Angle stays the same



Differentiated angle

c)

Single Step: Identify the angle and apply the chain rule process to a logarithmic function.

Here, we are differentiating a logarithmic function, so the angle is the argument (what we’re applying to). We first differentiate the angle, then the function of reciprocates the angle. The is a multiplier so we write that down first.

Here, the angle is , so we differentiate that using the chain rule for the function.

Angle stays the same



Differentiated angle

The angle differentiates to give . So,

Multiplier reciprocates the angle



Differentiated angle