Chain Rule - Power/Sine/Cosine/Exponential/Log Functions
Neil Trivedi
Teacher
Contents
Chain Rule
The chain rule is used to differentiate composite functions, which are in the form . We’ll derive a general formula for the chain rule.
is what we call the “angle” of our function, and we let that equal .
Let .
When we differentiate both functions, we get:
We want , so:
Chain Rule (notice how the 's cancel)
Angle stays the same
Differentiated angle
General Formula:
If
General Steps:
1) Differentiate the angle to get and write it in front.
2) Identify what type of function is and differentiate it according to our standard rules.
3) The angle never changes.
Power Functions
Example 1:
Differentiate .
Single Step: Identify the angle and apply the chain rule process to a power function.
Here, we are differentiating a power function, so the angle is what is inside the bracket. Here, it is , which differentiates to give . Then, for the power function, we bring down the power, knock off the power, then keep the angle the same.
Angle stays the same
Differentiated angle Differentiated power function
b)
Step 1: Rewrite the function.
Step 2: Identify the angle and apply the chain rule process to a power function.
As mentioned before, the angle of a power function is what’s inside the bracket.
Here, that is , which differentiates to give . Then, for the power function, we bring down the power and knock off the power, then keep the angle the same.
To note, the is a multiplier so we can simply write it first.
Multiplier Angle stays the same
Differentiated angle Differentiated power function
We can factorise out of our differentiated angle.
No answer provided.
Sine and Cosine
Formulae:
The results above can be derived using first principles (see the Derivative of Sine from First Principles note). How about a function like ? We can derive the results of using the chain rule.
Let . So, we have:
Differentiating both,
Using the chain rule,
Angle stays the same
Differentiated angle
1) Notice that the angle was differentiated first to give and written at the front.
2) differentiates to .
3) The angle didn’t change.
The result of can be similarly derived using the chain rule.
Formulae:
Example 2:
Differentiate
a)
Single Step: Identify the angle and apply the chain rule process to the sine function.
The angle is , which differentiates to , then differentiates to , then the angle stays the same.
Angle stays the same
Differentiated angle
b)
Step 1: Identify the type of function.
Although this looks like a trigonometric function, this is actually a power function, and we need to rewrite the function as .
Single Step: Identify the angle and apply the chain rule process to a power function.
The angle is , which differentiates to . For the power function, we bring down the power and knock off the power. The angle doesn’t change.
Also, to note, the is a multiplier so we can write it first.
So,
Multiplier Angle stays the same
Differentiated angle Differentiated power function
No answer provided.
Exponentials and Logarithms
Natural Exponentials
For exponential functions, the angle is the power. We differentiate the angle, then the function of stays the same in differentiation as well as the power (hence, the angle doesn’t change).
Natural Logarithms
For logarithmic functions, the angle is the argument (what we are taking log of). We differentiate the angle, then when we differentiate the function of ln, the angle is reciprocated.
To prove that , we’ll use the chain rule.
The angle of this function is , which differentiates to . Then, when we differentiate the function, the angle is reciprocated. So,
Differentiated angle When is differentiated,
the angle is reciprocated.
General Exponentials
Let's prove that:
First, write as a power of .
Notice how and are inverse functions and cancel so the equation still reads .
Using logarithmic rules, we can bring down the so that it’s outside the .
Apply the chain rule process for exponential functions. The angle is , which differentiates to (to note, is a constant, in the same way differentiates to ),
Angle stays the same
Differentiated angle
Note: The is known as our base.
We could similarly prove that .
Example 3:
Differentiate
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. The angle is , which differentiates to . The function of stays the same in differentiation as well as its power.
So,
Angle stays the same
Differentiated angle
b)
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 are taking log of). The angle is , which differentiates to . Then, when we differentiate the function of , the angle is reciprocated. The is a multiplier so we can write it first.
So,
When ln is differentiated,
Multiplier the angle is reciprocated.
Differentiated angle
Note: This is just an application of the concept. In this question we could have use our logarithmic rules to bring down the power of to get which directly differentiates to .
c)
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, which is , and that differentiates to . Then we use the fact that differentiates to .
We then multiply the
So,
Angle stays the same
Differentiated angle differentiates to
No answer provided.
Application
Example 4:
A curve has equation . Find the stationary points on the curve, .
Step 1: Differentiate the equation.
The angle is , which differentiates to , then differentiates to . The angle never changes.
Angle stays the same differentiates to
Differentiated angle
Step 2: The stationary points on the curve occur when the gradient is equal to , so we solve .
Now, we need to solve this trigonometric equation. We modify the range to read by multiplying all values by .
Find the PV. To note, you ensure you don’t divide by until you’ve found all values within the modified range.
(PV)
Trigonometric Functions | Finding the SV |
| PV |
| PV |
| PV |
Find the SV by working out PV.
PV (SV)
Both the PV and SV are in the modified range. If we add to the PV and SV, we’d get values that lie outside the modified range so we can stop here.
Note: To maintain accuracy, store the exact values of into the calculator and use them to find the values of . Then, we can round in our final answer.
Step 3: Substitute the values of we found in step 2 to find the corresponding
coordinates.
Therefore, the coordinates of the stationary points are, to significant figures, and .
No answer provided.
Challenging Questions