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How to Find the Derivative of $f(x)=\\sqrt{x^2+5x}$ | Step-by-Step Calculation
Mathematics
Grade 10 (Junior High School)
Question Content
Find the derivative of the function $f(x) = \\sqrt{x^2 + 5x}$ (the German text indicates the task: 'Sie einen Funktion' / 'unktion der Funktion' translates to 'Find the derivative of the function')
Correct Answer
$f'(x)=\\frac{2x+5}{2\\sqrt{x^2+5x}}$
Detailed Solution Steps
1
Step 1: Rewrite the function in exponential form to apply the power rule: $f(x)=(x^2+5x)^{\\frac{1}{2}}$
2
Step 2: Apply the chain rule: if $f(u)=u^{\\frac{1}{2}}$ and $u(x)=x^2+5x$, then $f'(x)=f'(u)\\cdot u'(x)$
3
Step 3: Calculate $f'(u)$ using the power rule $\\frac{d}{du}u^n=nu^{n-1}$: $f'(u)=\\frac{1}{2}u^{-\\frac{1}{2}}=\\frac{1}{2\\sqrt{u}}$
4
Step 4: Calculate $u'(x)$ using the sum and power rules: $u'(x)=\\frac{d}{dx}(x^2)+\\frac{d}{dx}(5x)=2x+5$
5
Step 5: Substitute back $u=x^2+5x$ and combine the results: $f'(x)=\\frac{1}{2\\sqrt{x^2+5x}}\\cdot(2x+5)=\\frac{2x+5}{2\\sqrt{x^2+5x}}$
Knowledge Points Involved
1
Chain Rule for Differentiation
A rule for differentiating composite functions, stating that if $y=f(u)$ and $u=g(x)$, then $\\frac{dy}{dx}=\\frac{dy}{du}\\cdot\\frac{du}{dx}$. Used when a function is nested inside another function, like a square root of a polynomial.
2
Power Rule for Differentiation
A basic differentiation rule: $\\frac{d}{dx}x^n=nx^{n-1}$ for any real number $n$. Applies to monomial terms, and can be extended to radical functions by rewriting them with fractional exponents.
3
Sum Rule for Differentiation
States that the derivative of a sum of functions is the sum of their derivatives: $\\frac{d}{dx}[g(x)+h(x)]=g'(x)+h'(x)$. Used to differentiate polynomial functions term-by-term.
4
Rewriting Radical Functions
Radical expressions can be rewritten using fractional exponents: $\\sqrt[k]{x^n}=x^{\\frac{n}{k}}$. This allows the use of the power rule to differentiate radical functions, which is simpler than working directly with radical notation.
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