# Calculate f'(x) given d/dx(f(3x^4))=6x^4?

• firegoalie33
In summary, to find f'(x) we can use the power rule for derivatives. We divide both sides by 3 and substitute x for 3x^4 to get f'(x) = 2x^3. f'(x) represents the derivative of the function f(x) with respect to x, while d/dx(f(x)) represents the derivative of the entire function f(x). The quotient rule can be used to find f'(x), but in this case, the power rule is more efficient. The chain rule can also be used by multiplying the derivative of the outer function by the derivative of the inner function. The product rule is not applicable in this scenario, as we are dealing with only one function, f
firegoalie33
I am told that d/dx(f(3x^4))=6x^4. I need to calculate f ' (x). i have tried like 3 different methods and have no idea how to do this.

Can you find f'(3x4)?

By the chain rule,
$$\frac{df(3x^4)}{dx}= f'(3x^4)(12x^3)$$

So you must have $f'(3x^4)(12x^3)= 6x^4$.

That let's you find $f'(3x^4)$ per Office Shredder's question. Now, if you let $u= 3x^4$, you can find f'(u).

## 1. How do I find f'(x) given d/dx(f(3x^4))=6x^4?

To find f'(x), we can use the power rule for derivatives. We know that d/dx(x^n) = nx^(n-1), so in this case, we have d/dx(f(3x^4)) = 6x^4. This means that f'(3x^4) = 6x^4. To find f'(x), we simply need to divide both sides by 3 and substitute x for 3x^4, giving us f'(x) = 2x^3.

## 2. What is the difference between f'(x) and d/dx(f(x))?

f'(x) represents the derivative of the function f(x) with respect to x. This means that it tells us the rate of change of f(x) at a specific point, or the slope of the tangent line at that point. On the other hand, d/dx(f(x)) represents the derivative of the entire function f(x), not just at a specific point. It tells us the general rate of change of the function at any point along its curve.

## 3. Can I use the quotient rule to find f'(x) instead of the power rule?

Yes, you can use the quotient rule to find f'(x) as well. However, in this specific case, the power rule is easier and more efficient to use. The quotient rule is typically used when the function is in the form of f(x)/g(x), where f(x) and g(x) are both functions of x.

## 4. How does the chain rule apply in this scenario?

The chain rule states that when finding the derivative of a composite function, we must multiply the derivative of the outer function by the derivative of the inner function. In this case, our composite function is f(3x^4), where the outer function is f(x) and the inner function is 3x^4. Therefore, the chain rule would be used to find f'(x) by multiplying the derivative of f(x) by the derivative of 3x^4, which is 12x^3.

## 5. Can I use the product rule to find f'(x) in this case?

No, the product rule is not applicable in this scenario because we are not dealing with a product of two functions. The product rule is used when finding the derivative of a function that is in the form of f(x)g(x), where f(x) and g(x) are both functions of x. In this case, we only have one function, f(x), and d/dx(f(3x^4)) is simply a notation for the derivative of that function with respect to x.

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