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- Find the derivative using the definition
- Integrate by partial fractions
- Product of Binomials with Common Term
- FOIL Method
- Integrate by substitution
- Integrate by parts
- Integrate using tabular integration
- Integrate by trigonometric substitution
- Weierstrass Substitution
- Prove from LHS (left-hand side)
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Find the derivative of $\frac{x^3}{1+x}$ using the definition. Apply the definition of the derivative: $\displaystyle f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$. The function $f(x)$ is the function we want to differentiate, which is $\frac{x^3}{1+x}$. Substituting $f(x+h)$ and $f(x)$ on the limit, we get
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$\lim_{h\to0}\left(\frac{\frac{\left(x+h\right)^3}{1+x+h}-\frac{x^3}{1+x}}{h}\right)$
Learn how to solve problems step by step online. Find the derivative of (x^3)/(1+x) using the definition. Find the derivative of \frac{x^3}{1+x} using the definition. Apply the definition of the derivative: \displaystyle f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}. The function f(x) is the function we want to differentiate, which is \frac{x^3}{1+x}. Substituting f(x+h) and f(x) on the limit, we get. Combine \frac{\left(x+h\right)^3}{1+x+h}-\frac{x^3}{1+x} in a single fraction. Combine \left(x+h\right)^3+\frac{-x^3\left(1+x+h\right)}{1+x} in a single fraction. Divide fractions \frac{\frac{\frac{-x^3\left(1+x+h\right)+\left(x+h\right)^3\left(1+x\right)}{1+x}}{1+x+h}}{h} with Keep, Change, Flip: \frac{a}{b}\div c=\frac{a}{b}\div\frac{c}{1}=\frac{a}{b}\times\frac{1}{c}=\frac{a}{b\cdot c}.
