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

Studdy Solution

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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