Table of Derivatives d d x ( c ) = 0 \frac{d}{dx}(c)=0 d x d ( c ) = 0 d d x [ c f ( x ) ] = c f ′ ( x ) \frac{d}{dx}[cf(x)]=cf^{\prime}(x) d x d [ c f ( x )] = c f ′ ( x ) d d x ( x n ) = n x n − 1 , for real numbers n \frac{d}{dx}(x^n)=nx^{n-1}\text{, for real numbers n} d x d ( x n ) = n x n − 1 , for real numbers n d d x [ f ( x ) + g ( x ) ] = f ′ ( x ) + g ′ ( x ) \frac{d}{dx}[f(x)+g(x)]=f^{\prime}(x)+g^{\prime}(x) d x d [ f ( x ) + g ( x )] = f ′ ( x ) + g ′ ( x ) d d x [ f ( x ) − g ( x ) ] = f ′ ( x ) − g ′ ( x ) \frac{d}{dx}[f(x)-g(x)]=f^{\prime}(x)-g^{\prime}(x) d x d [ f ( x ) − g ( x )] = f ′ ( x ) − g ′ ( x ) d d x [ f ( x ) g ( x ) ] = f ′ ( x ) g ( x ) + f ( x ) g ′ ( x ) \frac{d}{dx}[f(x)g(x)]=f^{\prime}(x)g(x)+f(x)g^{\prime}(x) d x d [ f ( x ) g ( x )] = f ′ ( x ) g ( x ) + f ( x ) g ′ ( x ) d d x [ f ( x ) g ( x ) ] = g ( x ) f ′ ( x ) − f ( x ) g ′ ( x ) [ g ( x ) ] 2 \frac{d}{dx}[\frac{f(x)}{g(x)}]=\frac{g(x)f^{\prime}(x)-f(x)g^{\prime}(x)}{[g(x)]^2} d x d [ g ( x ) f ( x ) ] = [ g ( x ) ] 2 g ( x ) f ′ ( x ) − f ( x ) g ′ ( x ) d d x [ f ( g ( x ) ) ] = f ′ ( g ( x ) ) ⋅ g ′ ( x ) \frac{d}{dx}[f(g(x))]=f^{\prime}(g(x))\cdot g^{\prime}(x) d x d [ f ( g ( x ))] = f ′ ( g ( x )) ⋅ g ′ ( x ) Trigonometric Functions d d x [ sin ( x ) ] = cos ( x ) \frac{d}{dx}[\sin(x)]=\cos(x) d x d [ sin ( x )] = cos ( x ) d d x [ cos ( x ) ] = − sin ( x ) \frac{d}{dx}[\cos(x)]=-\sin(x) d x d [ cos ( x )] = − sin ( x ) d d x [ tan ( x ) ] = sec 2 ( x ) \frac{d}{dx}[\tan(x)]=\sec^2(x) d x d [ tan ( x )] = sec 2 ( x ) d d x [ csc ( x ) ] = − csc ( x ) cot ( x ) \frac{d}{dx}[\csc(x)]=-\csc(x)\cot(x) d x d [ csc ( x )] = − csc ( x ) cot ( x ) d d x [ sec ( x ) ] = sec ( x ) tan ( x ) \frac{d}{dx}[\sec(x)]=\sec(x)\tan(x) d x d [ sec ( x )] = sec ( x ) tan ( x ) d d x [ cot ( x ) ] = − csc 2 ( x ) \frac{d}{dx}[\cot(x)]=-\csc^2(x) d x d [ cot ( x )] = − csc 2 ( x ) Inverse Trigonometric Functions d d x [ sin − 1 ( x ) ] = 1 1 − x 2 \frac{d}{dx}[\sin^{-1}(x)]=\frac{1}{\sqrt{1 - x^2}} d x d [ sin − 1 ( x )] = 1 − x 2 1 d d x [ cos − 1 ( x ) ] = − 1 1 − x 2 \frac{d}{dx}[\cos^{-1}(x)]=-\frac{1}{\sqrt{1 - x^2}} d x d [ cos − 1 ( x )] = − 1 − x 2 1 d d x [ tan − 1 ( x ) ] = 1 1 + x 2 \frac{d}{dx}[\tan^{-1}(x)]=\frac{1}{1 + x^2} d x d [ tan − 1 ( x )] = 1 + x 2 1 d d x [ csc − 1 ( x ) ] = − 1 ∣ x ∣ x 2 − 1 \frac{d}{dx}[\csc^{-1}(x)]=-\frac{1}{|x|\sqrt{x^2 - 1}} d x d [ csc − 1 ( x )] = − ∣ x ∣ x 2 − 1 1 d d x [ sec − 1 ( x ) ] = 1 ∣ x ∣ x 2 − 1 \frac{d}{dx}[\sec^{-1}(x)]=\frac{1}{|x|\sqrt{x^2 - 1}} d x d [ sec − 1 ( x )] = ∣ x ∣ x 2 − 1 1 d d x [ cot − 1 ( x ) ] = − 1 1 + x 2 \frac{d}{dx}[\cot^{-1}(x)]=-\frac{1}{1 + x^2} d x d [ cot − 1 ( x )] = − 1 + x 2 1 Exponential and Logarithmic Functions d d x ( e x ) = e x \frac{d}{dx}(e^x)=e^x d x d ( e x ) = e x d d x ( a x ) = a x ln ( a ) \frac{d}{dx}(a^x)=a^x\ln(a) d x d ( a x ) = a x ln ( a ) d d x [ ln ( ∣ x ∣ ) ] = 1 x \frac{d}{dx}[\ln(|x|)]=\frac{1}{x} d x d [ ln ( ∣ x ∣ )] = x 1 d d x [ log a ( x ) ] = 1 x ln ( a ) \frac{d}{dx}[\log_a(x)]=\frac{1}{x\ln(a)} d x d [ log a ( x )] = x l n ( a ) 1 Hyperbolic Functions d d x [ sinh ( x ) ] = cosh ( x ) \frac{d}{dx}[\sinh(x)]=\cosh(x) d x d [ sinh ( x )] = cosh ( x ) d d x [ cosh ( x ) ] = sinh ( x ) \frac{d}{dx}[\cosh(x)]=\sinh(x) d x d [ cosh ( x )] = sinh ( x ) d d x [ tanh ( x ) ] = sech 2 ( x ) \frac{d}{dx}[\tanh(x)]=\text{sech}^2(x) d x d [ tanh ( x )] = sech 2 ( x ) d d x [ csch ( x ) ] = − csch ( x ) coth ( x ) \frac{d}{dx}[\text{csch}(x)]=-\text{csch}(x)\coth(x) d x d [ csch ( x )] = − csch ( x ) coth ( x ) d d x [ sech ( x ) ] = − sech ( x ) tanh ( x ) \frac{d}{dx}[\text{sech}(x)]=-\text{sech}(x)\tanh(x) d x d [ sech ( x )] = − sech ( x ) tanh ( x ) d d x [ coth ( x ) ] = − csch 2 ( x ) \frac{d}{dx}[\coth(x)]=-\text{csch}^2(x) d x d [ coth ( x )] = − csch 2 ( x ) Inverse Hyperbolic Functions d d x [ sinh − 1 ( x ) ] = 1 x 2 + 1 \frac{d}{dx}[\sinh^{-1}(x)]=\frac{1}{\sqrt{x^2 + 1}} d x d [ sinh − 1 ( x )] = x 2 + 1 1 d d x [ cosh − 1 ( x ) ] = 1 x 2 − 1 , (x > 1) \frac{d}{dx}[\cosh^{-1}(x)]=\frac{1}{\sqrt{x^2 - 1}}\text{, (x > 1)} d x d [ cosh − 1 ( x )] = x 2 − 1 1 , (x > 1) d d x [ tanh − 1 ( x ) ] = 1 1 − x 2 , (|x| < 1) \frac{d}{dx}[\tanh^{-1}(x)]=\frac{1}{1 - x^2}\text{, (|x| < 1)} d x d [ tanh − 1 ( x )] = 1 − x 2 1 , (|x| < 1) d d x [ csch − 1 ( x ) ] = − 1 ∣ x ∣ 1 + x 2 , (x ≠ 0) \frac{d}{dx}[\text{csch}^{-1}(x)]=-\frac{1}{|x|\sqrt{1 + x^2}}\text{, (x}\not=\text{0)} d x d [ csch − 1 ( x )] = − ∣ x ∣ 1 + x 2 1 , (x = 0) d d x [ sech − 1 ( x ) ] = − 1 x 1 − x 2 , (0 < x < 1) \frac{d}{dx}[\text{sech}^{-1}(x)]=-\frac{1}{x\sqrt{1 - x^2}}\text{, (0 < x < 1)} d x d [ sech − 1 ( x )] = − x 1 − x 2 1 , (0 < x < 1) d d x [ coth − 1 ( x ) ] = 1 1 − x 2 , (|x| > 1) \frac{d}{dx}[\coth^{-1}(x)]=\frac{1}{1 - x^2}\text{, (|x| > 1)} d x d [ coth − 1 ( x )] = 1 − x 2 1 , (|x| > 1) Additional Notes If you're learning Calculus, you may have recognized several of these rules. Many are commonly used and have names associated with them, such as:
3 . Power Rule – Used to differentiate powers of x x x 6 . Product Rule – Used to differentiate the product of two functions.7 . Quotient Rule – Used to differentiate the ratio of two functions.8 . Chain Rule – Used to differentiate composite functions.Inverse trigonometric and hyperbolic functions (entries 15–20 and 31–36) also follow standard patterns, but are not typically given short names like the above rules.
