Solved on Feb 04, 2024

Find the derivative of $\sqrt{5 x^{2}-4}$ with respect to $x$ .

STEP 1

Assumptions
1. We need to differentiate the function $f(x) = \sqrt{5x^2 - 4}$ with respect to $x$ .
2. We will use the chain rule for differentiation, which states that if a function $y = f(g(x))$ , then the derivative $y'$ is $f'(g(x))g'(x)$ .
3. We will also use the power rule for differentiation, which states that if $f(x) = x^n$ , then the derivative $f'(x) = nx^{n-1}$ .

STEP 2

Rewrite the function in a form that is easier to differentiate using the power rule. The square root can be expressed as a power of $\frac{1}{2}$ .
$f(x) = (5x^2 - 4)^{\frac{1}{2}}$

STEP 3

Apply the chain rule for differentiation. Let $u = 5x^2 - 4$ , so that $f(x) = u^{\frac{1}{2}}$ . We need to find the derivatives $f'(u)$ and $u'(x)$ .

STEP 4

Differentiate $u$ with respect to $x$ using the power rule.
$u = 5x^2 - 4$ $u'(x) = \frac{d}{dx}(5x^2) - \frac{d}{dx}(4)$

STEP 5

Calculate the derivative of $5x^2$ with respect to $x$ .
$\frac{d}{dx}(5x^2) = 2 \cdot 5x^{2-1} = 10x$

STEP 6

Calculate the derivative of the constant $4$ with respect to $x$ .
$\frac{d}{dx}(4) = 0$

STEP 7

Combine the results from STEP_5 and STEP_6 to find $u'(x)$ .
$u'(x) = 10x - 0 = 10x$

STEP 8

Differentiate $f(u) = u^{\frac{1}{2}}$ with respect to $u$ using the power rule.
$f'(u) = \frac{1}{2}u^{\frac{1}{2} - 1} = \frac{1}{2}u^{-\frac{1}{2}}$

STEP 9

Substitute $u = 5x^2 - 4$ back into $f'(u)$ .
$f'(u) = \frac{1}{2}(5x^2 - 4)^{-\frac{1}{2}}$

STEP 10

Now, apply the chain rule by multiplying $f'(u)$ by $u'(x)$ to find the derivative of $f(x)$ with respect to $x$ .
$f'(x) = f'(u) \cdot u'(x)$

STEP 11

Substitute the expressions for $f'(u)$ and $u'(x)$ from STEP_9 and STEP_7 into the chain rule.
$f'(x) = \frac{1}{2}(5x^2 - 4)^{-\frac{1}{2}} \cdot 10x$

STEP 12

Simplify the expression by multiplying the constants outside the parentheses.
$f'(x) = \frac{10x}{2}(5x^2 - 4)^{-\frac{1}{2}}$

STEP 13

Further simplify the expression by dividing the constants.
$f'(x) = 5x(5x^2 - 4)^{-\frac{1}{2}}$
This is the derivative of $f(x) = \sqrt{5x^2 - 4}$ with respect to $x$ .

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Find the derivative of $\sqrt{5 x^{2}-4}$ with respect to $x$ .

STEP 1

STEP 2

Rewrite the function in a form that is easier to differentiate using the power rule. The square root can be expressed as a power of $\frac{1}{2}$ .
$f(x) = (5x^2 - 4)^{\frac{1}{2}}$

STEP 3

Apply the chain rule for differentiation. Let $u = 5x^2 - 4$ , so that $f(x) = u^{\frac{1}{2}}$ . We need to find the derivatives $f'(u)$ and $u'(x)$ .

STEP 4

Differentiate $u$ with respect to $x$ using the power rule.
$u = 5x^2 - 4$ $u'(x) = \frac{d}{dx}(5x^2) - \frac{d}{dx}(4)$

STEP 5

Calculate the derivative of $5x^2$ with respect to $x$ .
$\frac{d}{dx}(5x^2) = 2 \cdot 5x^{2-1} = 10x$

STEP 6

Calculate the derivative of the constant $4$ with respect to $x$ .
$\frac{d}{dx}(4) = 0$

STEP 7

Combine the results from STEP_5 and STEP_6 to find $u'(x)$ .
$u'(x) = 10x - 0 = 10x$

STEP 8

Differentiate $f(u) = u^{\frac{1}{2}}$ with respect to $u$ using the power rule.
$f'(u) = \frac{1}{2}u^{\frac{1}{2} - 1} = \frac{1}{2}u^{-\frac{1}{2}}$

STEP 9

Substitute $u = 5x^2 - 4$ back into $f'(u)$ .
$f'(u) = \frac{1}{2}(5x^2 - 4)^{-\frac{1}{2}}$

STEP 10

Now, apply the chain rule by multiplying $f'(u)$ by $u'(x)$ to find the derivative of $f(x)$ with respect to $x$ .
$f'(x) = f'(u) \cdot u'(x)$

STEP 11

Substitute the expressions for $f'(u)$ and $u'(x)$ from STEP_9 and STEP_7 into the chain rule.
$f'(x) = \frac{1}{2}(5x^2 - 4)^{-\frac{1}{2}} \cdot 10x$

STEP 12

Simplify the expression by multiplying the constants outside the parentheses.
$f'(x) = \frac{10x}{2}(5x^2 - 4)^{-\frac{1}{2}}$

STEP 13

Further simplify the expression by dividing the constants.
$f'(x) = 5x(5x^2 - 4)^{-\frac{1}{2}}$
This is the derivative of $f(x) = \sqrt{5x^2 - 4}$ with respect to $x$ .

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Find the derivative of 5x2−4\sqrt{5 x^{2}-4}5x2−4​ with respect to xxx.

STEP 1

STEP 2

Rewrite the function in a form that is easier to differentiate using the power rule. The square root can be expressed as a power of 12\frac{1}{2}21​.f(x)=(5x2−4)12f(x) = (5x^2 - 4)^{\frac{1}{2}}f(x)=(5x2−4)21​

STEP 3

Apply the chain rule for differentiation. Let u=5x2−4u = 5x^2 - 4u=5x2−4, so that f(x)=u12f(x) = u^{\frac{1}{2}}f(x)=u21​. We need to find the derivatives f′(u)f'(u)f′(u) and u′(x)u'(x)u′(x).

STEP 4

Differentiate uuu with respect to xxx using the power rule.u=5x2−4u = 5x^2 - 4u=5x2−4 u′(x)=ddx(5x2)−ddx(4)u'(x) = \frac{d}{dx}(5x^2) - \frac{d}{dx}(4)u′(x)=dxd​(5x2)−dxd​(4)

STEP 5

Calculate the derivative of 5x25x^25x2 with respect to xxx.ddx(5x2)=2⋅5x2−1=10x\frac{d}{dx}(5x^2) = 2 \cdot 5x^{2-1} = 10xdxd​(5x2)=2⋅5x2−1=10x

STEP 6

Calculate the derivative of the constant 444 with respect to xxx.ddx(4)=0\frac{d}{dx}(4) = 0dxd​(4)=0

STEP 7

Combine the results from STEP_5 and STEP_6 to find u′(x)u'(x)u′(x).u′(x)=10x−0=10xu'(x) = 10x - 0 = 10xu′(x)=10x−0=10x

STEP 8

Differentiate f(u)=u12f(u) = u^{\frac{1}{2}}f(u)=u21​ with respect to uuu using the power rule.f′(u)=12u12−1=12u−12f'(u) = \frac{1}{2}u^{\frac{1}{2} - 1} = \frac{1}{2}u^{-\frac{1}{2}}f′(u)=21​u21​−1=21​u−21​

STEP 9

Substitute u=5x2−4u = 5x^2 - 4u=5x2−4 back into f′(u)f'(u)f′(u).f′(u)=12(5x2−4)−12f'(u) = \frac{1}{2}(5x^2 - 4)^{-\frac{1}{2}}f′(u)=21​(5x2−4)−21​

STEP 10

Now, apply the chain rule by multiplying f′(u)f'(u)f′(u) by u′(x)u'(x)u′(x) to find the derivative of f(x)f(x)f(x) with respect to xxx.f′(x)=f′(u)⋅u′(x)f'(x) = f'(u) \cdot u'(x)f′(x)=f′(u)⋅u′(x)

STEP 11

Substitute the expressions for f′(u)f'(u)f′(u) and u′(x)u'(x)u′(x) from STEP_9 and STEP_7 into the chain rule.f′(x)=12(5x2−4)−12⋅10xf'(x) = \frac{1}{2}(5x^2 - 4)^{-\frac{1}{2}} \cdot 10xf′(x)=21​(5x2−4)−21​⋅10x

STEP 12

Simplify the expression by multiplying the constants outside the parentheses.f′(x)=10x2(5x2−4)−12f'(x) = \frac{10x}{2}(5x^2 - 4)^{-\frac{1}{2}}f′(x)=210x​(5x2−4)−21​

STEP 13

Further simplify the expression by dividing the constants.f′(x)=5x(5x2−4)−12f'(x) = 5x(5x^2 - 4)^{-\frac{1}{2}}f′(x)=5x(5x2−4)−21​This is the derivative of f(x)=5x2−4f(x) = \sqrt{5x^2 - 4}f(x)=5x2−4​ with respect to xxx.

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Find the derivative of $\sqrt{5 x^{2}-4}$ with respect to $x$ .

Rewrite the function in a form that is easier to differentiate using the power rule. The square root can be expressed as a power of $\frac{1}{2}$ .
$f(x) = (5x^2 - 4)^{\frac{1}{2}}$

Apply the chain rule for differentiation. Let $u = 5x^2 - 4$ , so that $f(x) = u^{\frac{1}{2}}$ . We need to find the derivatives $f'(u)$ and $u'(x)$ .

Differentiate $u$ with respect to $x$ using the power rule.
$u = 5x^2 - 4$ $u'(x) = \frac{d}{dx}(5x^2) - \frac{d}{dx}(4)$

Calculate the derivative of $5x^2$ with respect to $x$ .
$\frac{d}{dx}(5x^2) = 2 \cdot 5x^{2-1} = 10x$

Calculate the derivative of the constant $4$ with respect to $x$ .
$\frac{d}{dx}(4) = 0$

Combine the results from STEP_5 and STEP_6 to find $u'(x)$ .
$u'(x) = 10x - 0 = 10x$

Differentiate $f(u) = u^{\frac{1}{2}}$ with respect to $u$ using the power rule.
$f'(u) = \frac{1}{2}u^{\frac{1}{2} - 1} = \frac{1}{2}u^{-\frac{1}{2}}$

Substitute $u = 5x^2 - 4$ back into $f'(u)$ .
$f'(u) = \frac{1}{2}(5x^2 - 4)^{-\frac{1}{2}}$

Now, apply the chain rule by multiplying $f'(u)$ by $u'(x)$ to find the derivative of $f(x)$ with respect to $x$ .
$f'(x) = f'(u) \cdot u'(x)$

Substitute the expressions for $f'(u)$ and $u'(x)$ from STEP_9 and STEP_7 into the chain rule.
$f'(x) = \frac{1}{2}(5x^2 - 4)^{-\frac{1}{2}} \cdot 10x$

Simplify the expression by multiplying the constants outside the parentheses.
$f'(x) = \frac{10x}{2}(5x^2 - 4)^{-\frac{1}{2}}$

Further simplify the expression by dividing the constants.
$f'(x) = 5x(5x^2 - 4)^{-\frac{1}{2}}$
This is the derivative of $f(x) = \sqrt{5x^2 - 4}$ with respect to $x$ .