Answer:
[tex]\displaystyle y' = \frac{9 \bigg[ 6x^\big{\frac{9}{4}} \sqrt{x^3} \sin (x^3) - \sin (\sqrt{x}) \bigg] }{2x^\big{\frac{1}{4}}}[/tex]
General Formulas and Concepts:
Calculus
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Multiplied Constant]: [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]
Derivative Property [Addition/Subtraction]: [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]
Derivative Rule [Chain Rule]: [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]
Integration
Integration Rule [Fundamental Theorem of Calculus 2]: [tex]\displaystyle \frac{d}{dx}[\int\limits^x_a {f(t)} \, dt] = f(x)[/tex]
Integration Property [Multiplied Constant]: [tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]
Integration Property [Flipping Integral]: [tex]\displaystyle \int\limits^b_a {f(x)} \, dx = -\int\limits^a_b {f(x)} \, dx[/tex]
Integration Property [Splitting Integral]: [tex]\displaystyle \int\limits^c_a {f(x)} \, dx = \int\limits^b_a {f(x)} \, dx + \int\limits^c_b {f(x)} \, dx[/tex]
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle y = \int\limits^{x^3}_{\sqrt{x}} {9\sqrt{t} \sin (t)} \, dt[/tex]
Step 2: Differentiate
- [Integral] Rewrite [Integration Property - Multiplied Constant]: [tex]\displaystyle y = 9\int\limits^{x^3}_{\sqrt{x}} {\sqrt{t} \sin (t)} \, dt[/tex]
- [Integral] Rewrite [Integration Property - Splitting Integral]: [tex]\displaystyle y = 9 \bigg[ \int\limits^0_{\sqrt{x}} {\sqrt{t} \sin (t)} \, dt + \int\limits^{x^3}_0 {\sqrt{t} \sin (t)} \, dt \bigg][/tex]
- [1st Integral] Rewrite [Integration Property - Flipping Integral]: [tex]\displaystyle y = 9 \bigg[ -\int\limits^{\sqrt{x}}_0 {\sqrt{t} \sin (t)} \, dt + \int\limits^{x^3}_0 {\sqrt{t} \sin (t)} \, dt \bigg][/tex]
- Chain Rule [Integration Rule - Fundamental Theorem of Calculus 2]: [tex]\displaystyle y' = 9 \bigg[ - \bigg( {\sqrt{\sqrt{x}} \sin (\sqrt{x}) \bigg) \cdot \frac{d}{dx}[\sqrt{x}] + \bigg( \sqrt{x^3} \sin (x^3) \bigg) \cdot \frac{d}{dx}[x^3] \bigg][/tex]
- Basic Power Rule: [tex]\displaystyle y' = 9 \bigg[ - \bigg( {\sqrt{\sqrt{x}} \sin (\sqrt{x}) \bigg) \frac{1}{2\sqrt{x}} + \bigg( \sqrt{x^3} \sin (x^3) \bigg) \cdot 3x^2 \bigg][/tex]
- Simplify: [tex]\displaystyle y' = 9 \bigg[ \frac{- \bigg( x^\big{\frac{1}{4}} \sin (\sqrt{x}) \bigg)}{2\sqrt{x}} + \bigg( \sqrt{x^3} \sin (x^3) \bigg) \cdot 3x^2 \bigg][/tex]
- Rewrite: [tex]\displaystyle y' = \frac{9 \bigg[ 6x^\big{\frac{9}{4}} \sqrt{x^3} \sin (x^3) - \sin (\sqrt{x}) \bigg] }{2x^\big{\frac{1}{4}}}[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Integration
