Calculus Examples

$\int_{4}^{x^{2}} \sqrt{t} d t$

Step 1

Take the derivative of $\int_{4}^{x^{2}} \sqrt{t} d t$ with respect to $x$ using Fundamental Theorem of Calculus and the chain rule.

$\frac{d}{d x} [x^{2}] \sqrt{x^{2}}$

Step 2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x \sqrt{x^{2}}$

Step 3

Pull terms out from under the radical, assuming positive real numbers.

$2 x \cdot x$

Step 4

Raise $x$ to the power of $1$ .

$2 (x^{1} x)$

Step 5

Raise $x$ to the power of $1$ .

$2 (x^{1} x^{1})$

Step 6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x^{1 + 1}$

Step 7

Add $1$ and $1$ .

$2 x^{2}$