Calculus Examples

$\int_{x}^{x^{2}} f' (t) d t$

Split the integral into two integrals where $c$ is some value between $x$ and $x^{2}$ .

$\frac{d}{d x} [\int_{x}^{c} f' (t) d t + \int_{c}^{x^{2}} f' (t) d t]$

By the Sum Rule, the derivative of $\int_{x}^{c} f' (t) d t + \int_{c}^{x^{2}} f' (t) d t$ with respect to $x$ is $\frac{d}{d x} [\int_{x}^{c} f' (t) d t] + \frac{d}{d x} [\int_{c}^{x^{2}} f' (t) d t]$ .

$\frac{d}{d x} [\int_{x}^{c} f' (t) d t] + \frac{d}{d x} [\int_{c}^{x^{2}} f' (t) d t]$

Swap the bounds of integration.

$\frac{d}{d x} [- \int_{c}^{x} f' (t) d t] + \frac{d}{d x} [\int_{c}^{x^{2}} f' (t) d t]$

Take the derivative of $- \int_{c}^{x} f' (t) d t$ with respect to $x$ using Fundamental Theorem of Calculus.

$- f' (t) + \frac{d}{d x} [\int_{c}^{x^{2}} f' (t) d t]$

Take the derivative of $\int_{c}^{x^{2}} f' (t) d t$ with respect to $x$ using Fundamental Theorem of Calculus and the chain rule.

$- f' (t) + \frac{d}{d x} [x^{2}] f' (t)$

Differentiate using the Power Rule.

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

$- f' (t) + 2 x f' (t)$

Reorder terms.

$2 x f' (t) - f' (t)$