HOW TO FIND THE DERIVATIVE OF A COMPOSITE FUNCTION (6 STEPS)

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A composite function is a function that is written inside another.

For example, if f(x)=8x and g(x)=x²+1, then

$$f(g(x))=8(x^2+1)$$

$$f(g(x))=8x^2+8$$

Here, f(x) is called the outer function and g(x) is the inner function and we can say that f(x) depends on g(x)

To differentiate composite functions, we use the chain rule which is exemplified as:

$$\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}$$

That is, the derivative of a composite function is simply the derivative of the outer function multiplied by the derivative of the inner function.

HOW TO FIND THE DERIVATIVE OF A COMPOSITE FUNCTION

1. Given $y=f(x)$

2. Let the inner function be u

3. Obtain the derivative of the outer function

4. Obtain the derivative of the inner function

5. Multiply the derivative of the outer function by the derivative of the inner function obtained in steps 3 and 4 respectively

6. Finally, replace u with its original value.

Example

Differentiate $y=\ln(\tan x)$

Solution

Step 1: Given $y=f(x)$

$$y=\ln(\tan x)$$

Step 2: Let the inner function be u

The inner function is $\tan x$, hence $u=\tan x$ and $y=\ln(u)$

Step 3: Obtain the derivative of the outer function

$$y=\ln(u)$$

$$\frac{dy}{du}=\frac{1}{u}$$

Step 4: Obtain the derivative of the inner function

The inner function is $\tan x$ and because $u=\tan x$

$$\frac{du}{dx}=\sec^2x$$

Step 5: Multiply the derivative of the outer function by the derivative of the inner function

$$\frac{dy}{dx}=\frac{1}{u}\times\sec^2x$$

$$\frac{dy}{dx}=\frac{sec^2x}{u}$$

Step 6: replace u with its original value.

If $u=\tan x$, then

$$\frac{dy}{dx}=\frac{sec^2x}{\tan x }$$

In the next post, you will see more examples that will solidify your knowledge on the differentiation of composite functions

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