# How Do I Find the Derivative of a Complex Function in Calculus?

• m0286
In summary, the conversation is about finding the minimum time for a calculus question, but the person is stuck on finding the derivative of a given function. The expert provides a simplified version of the function and explains the steps to find the derivative using the chain rule. They also suggest thinking of square roots as powers for easier differentiation.
m0286
I am trying to do a question for calculus. I am supposed to find a minimum time for something.. I know how to do the question but I am stuck on how to find the derivative of this function it really confuses me.

(sq. root 1^2+x^2)/3 + 3-x/5
I need to find dt/dx... and I am clueless on how to find this?
THANKS TONS!

$$d\sqrt{Y}=\frac{dY}{2\sqrt{Y}}$$ Now for Y = A+B, the rule is:

Y'=A'+B'. In the case of a form such as S/T, you can employ the rule:

d(S/T) =$$\frac{S'T-ST'}{T^2}$$

However, in the problem above we don't need to use that since: d(X/3) is just X'/3. Again in the case

$$\sqrt{\frac{1+x^2}{3}}=\frac{\sqrt{1+x^2}}{\sqrt3}$$ which is simpler to work with.

Last edited:
To differentiate

$$t= \frac{\sqrt{1 + x^2}}{3}$$

put $u=1 + x^2$, then use the chain rule:

$$\frac{dt}{dx} = \frac{dt}{du} \frac{du}{dx}$$

Last edited:
Or pulling out constants might make it seem simpler.

$$\frac{1}{3}*\frac{d\sqrt{1+x^2}}{dx}$$ etc...

Most people find it easier to think of "square root" as "1/2 power". That is,
To find the derivative of $$\frac{\sqrt{1+x^2}}{3} + 3- \frac{x}{5}[/itex], write it as [tex]\frac{1}{3}(1+ x^2)^{\frac{1}{2}}+ 3- \frac{x}{5}[/itex] Then the derivative is [itex]\frac{1}{3}\frac{1}{2}(1+ x^2)^{-\frac{1}{2}}{2x}- \frac{1}{5}$$

Last edited by a moderator:
You Guys Are Awesome Thanks Sooo Much!

## 1. What is the definition of a derivative?

The derivative of a function is the slope of the tangent line at a specific point on the function's graph. It represents the instantaneous rate of change of the function at that point.

## 2. How do you find the derivative of a polynomial function?

To find the derivative of a polynomial function, you can use the power rule, which states that the derivative of x^n is nx^(n-1). For example, the derivative of 3x^2 would be 6x.

## 3. What is the chain rule and when is it used?

The chain rule is a formula used to find the derivative of a composite function. It states that the derivative of f(g(x)) is equal to f'(g(x)) * g'(x). It is used when the function has multiple layers of functions nested within each other.

## 4. Can you find the derivative of a trigonometric function?

Yes, the derivative of a trigonometric function can be found using the appropriate trigonometric identities and the chain rule. For example, the derivative of sin(x) would be cos(x).

## 5. Why is finding the derivative important in mathematics and science?

Finding the derivative is important in mathematics and science because it allows us to analyze and understand the behavior of functions, such as their rate of change and direction of movement. It is also essential in applications such as optimization, physics, and engineering.

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