Find the indicated derivative. f ( x ) = 6 x 4 − 4 3 f(x)=6x^4-4^3 f ( x ) = 6 x 4 − 4 3 f ′ ( x ) = f^{\prime}\left(x\right)= f ′ ( x ) =

In this problem, we're asked to find the derivative f prime of x of our function f of x, which is 6x to the power of 4 minus 4 cubed. So let's go ahead and get started here. Now, again, I'm going to fully break this one down using all of the rules that we know. So let's go ahead and jump right in. Now here for my first term, I had that 6x to the power of 4. Now in thinking about this derivative, since I have a constant here, I know that I can really just pull that constant out of the derivative. So this derivative is 6 times derivative d d x of x to the power of 4, effectively just pulling that 6 out to the front. Then I'm subtracting the derivative d d x of 4 cubed. Now let's go ahead and actually take these derivatives. So 6 times the d d x of x to the power of 4. In order to take this derivative here, I know that I need to use the power rule. Now my constant here, 6, is going to stay as is. And when using the power rule, I'm gonna pull that 4 out to the front and then decrease that exponent by 1. So the derivative of x to the power of 4 is going to be 4x cubed. Having decreased that exponent by 1, 4 minus 1 gives me my new power of 3. Now here, I'm going to subtract the derivative of 4 cubed. Now a common mistake here, because this is 4 to the power of something, is to want to use the power rule. But let's think about what 4 cubed is. 4 cubed is just 4 times 4 times 4. That's just a constant. So we don't need to use the power rule here because there is no variable. This is just a constant, and the derivative of any constant is just 0. So here, we're effectively subtracting 0, which is doing nothing to our function. So we can further simplify this. 6 times 4 is 24. This is 24 x cubed for the derivative f prime of x. And we are all done here. Let me know if you have any questions and let's continue practicing.

3. Techniques of Differentiation

Basic Rules of Differentiation

Business Calculus

3. Techniques of Differentiation

Basic Rules of Differentiation

Struggling with Business Calculus?

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Multiple Choice

Find the indicated derivative.
$f(x)=6x^4-4^3$
$f^{$\prime$}$\left$(x$\right$)=$

$4x^3$

$24x^3$

$4x^3-1$

$24x^3-1$

Verified step by step guidance

Step 1: Identify the function to differentiate. The given function is f(x) = 6x^4 - 4^3. Note that the term 4^3 is a constant.

Step 2: Apply the power rule to differentiate the term 6x^4. The power rule states that if f(x) = ax^n, then f'(x) = n * ax^(n-1).

Step 3: Differentiate the constant term -4^3. The derivative of any constant is 0.

Step 4: Combine the results from Step 2 and Step 3. The derivative of 6x^4 is 24x^3, and the derivative of -4^3 is 0.

Step 5: Simplify the expression to get the final derivative. The result is f'(x) = 24x^3.

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Basic Rules of Differentiation practice set

Find the indicated derivative.
$f(x)=6x^4-4^3$
$f^{\(\prime\)}\(\left\)(x\(\right\))=$