Find the derivative of the function.f(x)=(5x2−3x)f(x)=\sqrt{(5x^2-3x)}f(x)=(5x2−3x)

10 x − 3 2 5 x 2 − 3 x \frac{10x-3}{2\sqrt{5x^2-3x}}

Find the derivative of the function. f ( x ) = ( 5 x 2 − 3 x ) f(x)=\sqrt{(5x^2-3x)} f ( x ) = ( 5 x 2 − 3 x ) <path d="M263,681c0.7,0,18,39.7,52,119c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120c340,-704.7,510.7,-1060.3,512,-1067l0 -0c4.7,-7.3,11,-11,19,-11H40000v40H1012.3s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60zM1001 80h400000v40h-400000z">

In this problem, we're asked to find the derivative of the function f of x equals the square root of 5x squared minus 3x. So here I have a function inside of another function. I have the function 5x squared minus 3x underneath that square root. That's my outside function. That's gonna be a little bit easier to recognize if we rewrite this using a power because we know that the square root can be rewritten as to the power of 1 half. So I can rewrite this function f of x as f of x equals 5x squared minus 3x to the power of 1 half. So then I have my inside function here, this 5x squared minus 3x, And I have my outside function. That's that to the power of 1 half. Remember that to use the chain rule here, that's what we're going to be using because we have a function inside of a function, we need to start from the outside and work our way in. So here our outside function is that to the power of 1 half. And we're really just going to be treating that stuff inside as though it is a variable. Now here we're going to go ahead and apply the power rule. So to find our derivative here, f prime of x, we want to go ahead and use the power rule pulling that one half out to the front here. So I have a one half. I'm going to treat all that stuff inside as though it's variable. So I'm going to keep it the same, 5x squared minus 3x. And all of this gets raised to the power of 1 half minus 1, which is negative one half. Now we're not done yet because we need to work our way inside now taking the derivative of my inside function. Now here, I'm going to again use the power rule, pull that 2 out to the front, decrease that power by 1. That's going to give me 10x. And then that minus 3x, the derivative of a constant times x, is just going to be that constant. So this becomes a minus 3. Now here, we can rewrite this in a little bit more compact way because we know that to the power of negative one half means that we're dividing by the square root of that. So we can fully rewrite this final derivative, f prime of x is equal to 10x minus 3 divided by 2, because remember we have that 1 half times the square root of 5x squared minus 3x. And this is our final derivative, which we found using the chain rule. Let me know if you have any questions here, and let's continue to get more practice. I'll see you in the next video.

Find the derivative of the function. f ( x ) = ( 5 x 2 − 3 x ) f(x)=\sqrt{(5x^2-3x)} f ( x ) = ( 5 x 2 − 3 x ) <path d="M263,681c0.7,0,18,39.7,52,119 c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120 c340,-704.7,510.7,-1060.3,512,-1067 l0 -0 c4.7,-7.3,11,-11,19,-11 H40000v40H1012.3 s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232 c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1 s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26 c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z M1001 80h400000v40h-400000z">

10 x − 3 2 5 x 2 − 3 x \frac{10x-3}{2\sqrt{5x^2-3x}}

1 2 10 x − 3 \frac12\sqrt{10x-3} 2 1 10 x − 3 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

1 2 10 x − 3 \frac{1}{2\sqrt{10x-3}} 2 10 x − 3 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> 1

1 2 5 x 2 − 3 \frac{1}{2\sqrt{5x^2-3}} 2 5 x 2 − 3 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> 1

3. Techniques of Differentiation

The Chain Rule

Calculus

3. Techniques of Differentiation

The Chain Rule

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Multiple Choice

Find the derivative of the function.
$f(x)=\sqrt{(5x^2-3x)}$

$\frac12\sqrt{10x-3}$

$\frac{1}{2\sqrt{10x-3}}$

$\frac{10 x - 3}{2 \sqrt{5 x^{2} - 3 x}}$

$\frac{1}{2\sqrt{5x^2-3}}$

Verified step by step guidance

Identify the function for which you need to find the derivative: $ f(x) = \sqrt{5x^2 - 3x} $. This is a composition of functions, where the outer function is a square root and the inner function is a quadratic expression.

Apply the chain rule for differentiation. The chain rule states that if you have a composite function $ f(g(x)) $, its derivative is $ f'(g(x)) \cdot g'(x) $. Here, let $ u = 5x^2 - 3x $, so $ f(x) = \sqrt{u} $.

Differentiate the outer function $ \sqrt{u} $ with respect to $ u $. The derivative of $ \sqrt{u} $ is $ \frac{1}{2\sqrt{u}} $.

Differentiate the inner function $ u = 5x^2 - 3x $ with respect to $ x $. The derivative of $ 5x^2 $ is $ 10x $ and the derivative of $ -3x $ is $ -3 $, so $ u' = 10x - 3 $.

Combine the results using the chain rule: $ f'(x) = \frac{1}{2\sqrt{5x^2 - 3x}} \cdot (10x - 3) $. Simplify this expression to get the final derivative: $ \frac{10x - 3}{2\sqrt{5x^2 - 3x}} $.

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