5.3 The Fundamental Theorem of Calculus/15: Difference between revisions
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\begin{align} | \begin{align} | ||
\frac{d}{dx}\\ | \frac{d}{dx}= \int_{0}^{tan(x)}\sqrt{t+\sqrt t}\,dt\\ | ||
&= \sec^{2}(x)\cdot\sqrt{tan(x)+\sqrt tan(x)})-0\cdot\sqrt{0+\sqrt 0}\\ | &= \sec^{2}(x)\cdot\sqrt{tan(x)+\sqrt tan(x)})-0\cdot\sqrt{0+\sqrt 0}\\ | ||
&= \sec^{2}(x)\cdot\sqrt{tan(x)+\sqrt tan(x)})\\ | &= \sec^{2}(x)\cdot\sqrt{tan(x)+\sqrt tan(x)})\\ | ||
Revision as of 19:54, 25 August 2022
Use part 1 of the FTC to find the derivative of the function: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle y=\int_{0}^{tan(x)}\sqrt{t+\sqrt t}\,dt}
FTC 1: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx}\int_{a(x)}^{b(x)}f(t)\,dt=b^\prime{(x)}\cdot\,f(b(x))-\,a^\prime{(x)}\cdot\,f(a(x))}
Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}{\frac {d}{dx}}=\int _{0}^{tan(x)}{\sqrt {t+{\sqrt {t}}}}\,dt\\&=\sec ^{2}(x)\cdot {\sqrt {tan(x)+{\sqrt {t}}an(x)}})-0\cdot {\sqrt {0+{\sqrt {0}}}}\\&=\sec ^{2}(x)\cdot {\sqrt {tan(x)+{\sqrt {t}}an(x)}})\\\end{aligned}}}
In this problem Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a^\prime{(x)}= 0}
, so when it is multiplied by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(a(x))}
it will result in 0 and doesn't need to be added.