Derivative of \(\sqrt{x}\)
We want to find the derivative of \(\sqrt{x}\) with respect to \(x\).
| Math | Explanation |
|---|---|
| \(\sqrt{x} = x^{1/2}\) | The square root of \(x\) is equivalent to raising \(x\) to the power of \(1/2\). |
| \(\frac{1}{2} \cdot x^{(1/2 – 1)}\) | Apply the power rule for derivatives and subtract 1 from the exponent. |
| \(\frac{1}{2} \cdot x^{-1/2}\) | The result of subtracting 1 from \(1/2\) is \(-1/2\). |
| \(\frac{1}{2} \cdot \frac{1}{x^{1/2}}\) | A negative exponent means taking the reciprocal. |
| \(\frac{1}{2\sqrt{x}}\) | The \(1/2\) exponent means taking the square root. |
Final Result:
The derivative of \(\sqrt{x}\) with respect to \(x\) is \(\frac{1}{2\sqrt{x}}\).
Derivative of \(\sqrt[3]{x}\)
We want to find the derivative of \(\sqrt[3]{x}\) with respect to \(x\).
| Math | Explanation |
|---|---|
| \(\sqrt[3]{x} = x^{1/3}\) | The cube root of \(x\) is equivalent to raising \(x\) to the power of \(1/3\). |
| \(\frac{1}{3} \cdot x^{(1/3 – 1)}\) | Apply the power rule for derivatives and subtract 1 from the exponent. |
| \(\frac{1}{3} \cdot x^{-2/3}\) | The result of subtracting 1 from \(1/3\) is \(-2/3\). |
| \(\frac{1}{3} \cdot \frac{1}{x^{2/3}}\) | A negative exponent means taking the reciprocal. |
| \(\frac{1}{3\sqrt[3]{x^2}}\) | The \(2/3\) exponent means taking the cube root of \(x^2\). |
Final Result:
The derivative of \(\sqrt[3]{x}\) with respect to \(x\) is \(\frac{1}{3\sqrt[3]{x^2}}\).
Derivative of \(\sqrt[4]{x}\)
We want to find the derivative of \(\sqrt[4]{x}\) with respect to \(x\).
| Math | Explanation |
|---|---|
| \(\sqrt[4]{x} = x^{1/4}\) | The fourth root of \(x\) is equivalent to raising \(x\) to the power of \(1/4\). |
| \(\frac{1}{4} \cdot x^{(1/4 – 1)}\) | Apply the power rule for derivatives and subtract 1 from the exponent. |
| \(\frac{1}{4} \cdot x^{-3/4}\) | The result of subtracting 1 from \(1/4\) is \(-3/4\). |
| \(\frac{1}{4} \cdot \frac{1}{x^{3/4}}\) | A negative exponent means taking the reciprocal. |
| \(\frac{1}{4\sqrt[4]{x^3}}\) | The \(3/4\) exponent means taking the fourth root of \(x^3\). |
Final Result:
The derivative of \(\sqrt[4]{x}\) with respect to \(x\) is \(\frac{1}{4\sqrt[4]{x^3}}\).
Derivative of the \(n\)-th Root of \(a\)
We want to find the derivative of the \(n\)-th root of \(a\) with respect to \(x\).
| Math | Explanation |
|---|---|
| \(\sqrt[n]{a} = a^{1/n}\) | The \(n\)-th root of \(a\) is equivalent to raising \(a\) to the power of \(1/n\). |
| \(\frac{d}{dx} a^{1/n} = \frac{1}{n} \cdot a^{(1/n – 1)}\) | Apply the power rule for derivatives and subtract 1 from the exponent. |
| \(\frac{1}{n} \cdot a^{(1/n – 1)} = \frac{1}{n} \cdot a^{-1/n}\) | The result of subtracting 1 from \(1/n\) is \(-1/n\). |
| \(\frac{1}{n} \cdot a^{-1/n} = \frac{1}{n\sqrt[n]{a}}\) | A negative exponent means taking the reciprocal, and the \(1/n\) exponent means taking the \(n\)-th root. |
Final Result:
The derivative of the \(n\)-th root of \(a\) with respect to \(x\) is \(\frac{1}{n\sqrt[n]{a}}\).