# Learning Calculus: Chain Rule and Derivatives

• MotoPayton
In summary, the derivative of a function can be found by taking the derivative of the exponential function. The chain rule can be used to find the derivative of a function when you know the function's form.
MotoPayton
I am currently learning calculus and just had my lecture on the chain rule.

I noticed that we haven't learned how to take the derivative of a function like 2^2+x or 3^4+x.
Any example works.. Is this something I will learn later as I progress through calculus or what?

MotoPayton said:
I am currently learning calculus and just had my lecture on the chain rule.

I noticed that we haven't learned how to take the derivative of a function like 2^2+x or 3^4+x.
Any example works.. Is this something I will learn later as I progress through calculus or what?

Assuming you meant what you wrote, these are very simple to differentiate. 2^2 + x = 4 + x, and its derivative is 1.

3^4 + x = 81 + x, and its derivative is also 1.

Now, assuming you meant 2^(2 + x) = 22 + x, and 3^(4 + x) = 34 + x, these functions can be differentiated when you know the chain rule form of the derivative of the exponential function.

chain rule : F'(x) = f(g(x)) g'(x)

example: take f(x) = 2sin(x) .....f'(x)=2sin(x)cos(x)

Not quite right. The derivative of 2^x is not just 2^x, you need to proceed as follows:
$$\frac{d}{dx} 2^x = \frac{d}{dx} e^{x log2} = \frac{d}{dx} (e^x)^{log2} = e^x log2*(e^x)^{log2-1} = log2*2^x$$

So:
$$\frac{d}{dx} 2^{sin(x)} = log2*cos(x)*2^{sin(x)}$$

my bad. not even a good example.

Generally, for any positive a, the derivative of $a^x$, with respect to x, is $(ln a)a^x$. Of course, if a= e, ln(a)= ln(e)= 1.

So finish this off ...
Derivative of 2^(2 +x) = (Ln 2) [2^(2+x)] (2+x)' = (Ln 2) [2^(2+x)]
and
Derivative of 3^(4 +x) = (Ln 3) [3^(4+x)] (4+x)' = (Ln 3) [3^(4+x)]

Just for fun checkout Derivative of 2^(2+x^2)
http://www.wolframalpha.com/input/?i=derivative+2^%282+%2B+x^2%29+dx
Click on "Show Steps" to see the solution

## 1. What is the chain rule in calculus?

The chain rule is a formula used in calculus to find the derivative of composite functions. It allows us to calculate the rate of change of a function composed of two or more functions by breaking it down into simpler parts.

## 2. Why is the chain rule important in calculus?

The chain rule is important because it allows us to find the derivative of more complex functions, which are often used to model real-world phenomena. It is a fundamental tool in many areas of mathematics, physics, and engineering.

## 3. How do you apply the chain rule in calculus?

To apply the chain rule, you need to identify the composite function and then use the chain rule formula to find its derivative. The formula is: (d/dx)(f(g(x))) = f'(g(x)) * g'(x), where f'(x) is the derivative of the outer function and g'(x) is the derivative of the inner function.

## 4. What are some common mistakes when using the chain rule in calculus?

Some common mistakes when using the chain rule include forgetting to apply the chain rule or using it incorrectly, not simplifying the function before taking the derivative, and not properly identifying the inner and outer functions. It is important to carefully follow the steps and practice regularly to avoid these mistakes.

## 5. How can I improve my understanding of the chain rule and derivatives in calculus?

To improve your understanding of the chain rule and derivatives in calculus, it is important to practice solving problems and working through examples. It can also be helpful to seek out additional resources, such as online tutorials or textbooks, and to seek help from a teacher or tutor if needed. Visual aids, such as graphs and diagrams, can also aid in understanding the concept.

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