General Derivative Formulas

1. [latex]\frac{d}{dx}(c)=0[/latex]

2. [latex]\frac{d}{dx}(f(x)+g(x))={f}^{\prime }(x)+{g}^{\prime }(x)[/latex]

3. [latex]\frac{d}{dx}(f(x)g(x))={f}^{\prime }(x)g(x)+f(x){g}^{\prime }(x)[/latex]

4. [latex]\frac{d}{dx}({x}^{n})=n{x}^{n-1},\text{for real numbers } n[/latex]

5. [latex]\frac{d}{dx}(cf(x))=c{f}^{\prime }(x)[/latex]

6. [latex]\frac{d}{dx}(f(x)-g(x))={f}^{\prime }(x)-{g}^{\prime }(x)[/latex]

7. [latex]\frac{d}{dx}(\frac{f(x)}{g(x)})=\frac{g(x){f}^{\prime }(x)-f(x){g}^{\prime }(x)}{{(g(x))}^{2}}[/latex]

8. [latex]\frac{d}{dx}\left[f(g(x))\right]={f}^{\prime }(g(x))\cdot{g}^{\prime }(x)[/latex]

Exponential and Logarithmic Derivatives

9. [latex]\frac{d}{dx}({e}^{x})={e}^{x}[/latex]

10. [latex]\frac{d}{dx}(\text{ln}|x|)=\frac{1}{x}[/latex]

11. [latex]\frac{d}{dx}({b}^{x})={b}^{x}\text{ln}b[/latex]

12. [latex]\frac{d}{dx}({\text{log}}_{b}x)=\frac{1}{x\text{ln}b}[/latex]

Basic Integrals

1. [latex]\int {u}^{n}du=\frac{{u}^{n+1}}{n+1}+C,n\ne -1[/latex]

2. [latex]\int \frac{du}{u}=\text{ln}|u|+C[/latex]

3. [latex]\int {e}^{u}du={e}^{u}+C[/latex]

Exponential and Logarithmic Integrals

4. [latex]\int u{e}^{au}du=\frac{1}{{a}^{2}}(au-1){e}^{au}+C[/latex]

5. [latex]\int {u}^{n}{e}^{au}du=\frac{1}{a}{u}^{n}{e}^{au}-\frac{n}{a}\int {u}^{n-1}{e}^{au}du[/latex]

6. [latex]\int \text{ln}udu=u\text{ln}u-u+C[/latex]

7. [latex]\int {u}^{n}\text{ln}udu=\frac{{u}^{n+1}}{{(n+1)}^{2}}\left[(n+1)\text{ln}u-1\right]+C[/latex]

8. [latex]\int \frac{1}{u\text{ln}u}du=\text{ln}|\text{ln}u|+C[/latex]