Find the indicated derivative.y=52x5+2x3+6x−√34y=\frac52x^5+2x^3+6x-\frac{\surd3}{4}y=25x5+2x3+6x−4√3y′=y^{\prime}=y′=

25 2 x 4 + 6 x 2 + 6 \frac{25}{2}x^4+6x^2+6 2 25 x 4 + 6 x 2 + 6

Find the indicated derivative. y = 5 2 x 5 + 2 x 3 + 6 x − √ 3 4 y=\frac52x^5+2x^3+6x-\frac{\surd3}{4} y = 2 5 x 5 + 2 x 3 + 6 x − 4 √3 y ′ = y^{\prime}= y ′ =

this problem, we're asked to find the derivative y prime of the function y equals 5 halves x to the power of 5 plus 2 x cubed plus 6 x minus root 3 over 4. So let's go ahead and jump in here. Now so far, I've been breaking down these derivatives fully based on all of the rules for derivatives that we've learned. But here, I'm gonna start to pick up the pace a little bit because more often as you get better at derivatives, you're gonna be able to take this derivative all at once. So let's go ahead and give that a try here. Now remember that we're effectively just taking the derivative of each of our individual terms and adding them together, but we can easily do that all at once. Now let's take a look at our first term here. We have 5 halves x to the power of 5. Now I know here that I wanna use the power rule because I have x raised to a power. Now in order to use the power rule, I need to pull that 5 out to the front and then decrease that power by 1. Now since I already have that constant there, 5 halves, I need to make sure that I'm multiplying that constant, five halves, by that 5 that I'm pulling out to the front using the power rule. So this is really 5 times 5 over 2. Now that gives me 25 over 2 times x to the power of 5 minus 1, which is just 4. Now let's move on to our next term. Here we have 2x cubed. We again wanna use the power rule here. So we wanna pull that 3 out to the front and then decrease that exponent by 1. Now, again, here, I already have that constant 2, so I need to make sure that I'm multiplying that 2 times the 3 that I'm pulling out to the front using the power rule. Now 2 times 3 gives me 6, so that's the constant that I have here. And this is times x to the power of 3 minus 1, which is just 2. Now for this next term, 6x, if I take the derivative of 6x, I'm taking the derivative of or I'm taking 6, my constant, and multiplying it by the derivative of x, which we know is just 1. So if I have a constant times x, just know that the derivative of that is always just going to be that constant by itself. So this ends up being just 6. Now for my final term, I'm subtracting root 3 over 4. Now looking at this, because we have a couple of different things going on here, you may be tempted to try to use something like the Power Rule. But you don't need to do that here because root 3 over 4 is just a number. It's just a constant. And the derivative of any constant, no matter what it looks like, no matter if it's a fraction or a square root or anything of the sort, it's always going to be 0 so we don't even have to add that term on the end here. And we have our final derivative, 25 over 2x to the power of 4 plus 6x squared plus 6. Let me know if you have any questions here and keep on practicing.

Find the indicated derivative. y = 5 2 x 5 + 2 x 3 + 6 x − √ 3 4 y=\frac52x^5+2x^3+6x-\frac{\surd3}{4} y = 2 5 x 5 + 2 x 3 + 6 x − 4 √3 y ′ = y^{\prime}= y ′ =

25 2 x 4 + 6 x 2 + 6 \frac{25}{2}x^4+6x^2+6 2 25 x 4 + 6 x 2 + 6

5 x 4 + 3 x 2 + 6 5x^4+3x^2+6 5 x 4 + 3 x 2 + 6

25 2 x 4 + 6 x 2 + 5 \frac{25}{2}x^4+6x^2+5 2 25 x 4 + 6 x 2 + 5

25 2 x 5 + 6 x 3 + 6 x − 1 \frac{25}{2}x^5+6x^3+6x-1 2 25 x 5 + 6 x 3 + 6 x − 1

3. Techniques of Differentiation

Basic Rules of Differentiation

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3. Techniques of Differentiation

Basic Rules of Differentiation

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Multiple Choice

Find the indicated derivative.
$y=\frac52x^5+2x^3+6x-\frac{\surd3}{4}$
$y^{\prime}=$

$5x^4+3x^2+6$

$\frac{25}{2}x^4+6x^2+5$

$\frac{25}{2}x^5+6x^3+6x-1$

$\frac{25}{2}x^4+6x^2+6$

Verified step by step guidance

Step 1: Identify the function to differentiate. The given function is y = (5/2)x^5 + 2x^3 + 6x - (√3)/4. This is a polynomial function, and we will apply the power rule to each term individually.

Step 2: Recall the power rule for differentiation. The power rule states that if y = ax^n, then y' = n * ax^(n-1). We will apply this rule to each term of the function.

Step 3: Differentiate the first term (5/2)x^5. Using the power rule, the derivative is (5/2) * 5 * x^(5-1) = (25/2)x^4.

Step 4: Differentiate the second term 2x^3. Using the power rule, the derivative is 3 * 2 * x^(3-1) = 6x^2. Similarly, differentiate the third term 6x, which gives 6, and the constant term -(√3)/4, which has a derivative of 0 since the derivative of a constant is always 0.

Step 5: Combine all the derivatives. Adding the results from each term, the derivative of the function is y' = (25/2)x^4 + 6x^2 + 6.

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