Derivative (mathematics) – simple english wikipedia, the free encyclopedia gas 4 less

The derivative of y with respect to x is defined as the change in y over the change in x, as the distance between x 0 {\displaystyle x_{0}} and x 1 {\displaystyle x_{1}} becomes infinitely small ( infinitesimal). In mathematical terms, f ′ ( a ) = lim h → 0 f ( a + h ) − f ( a ) h {\displaystyle f'(a)=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}}

That is, as the distance between the two x points (h) becomes closer to zero, the slope of the line between them comes closer to resembling a tangent line. Derivatives of functions [ change | change source ] Linear functions [ change | change source ]

When y modifies x’s number by adding or subtracting a constant value, the slope is still one because the change in x and y do not change if the graph is shifted up or down. That is, the slope is still 1 throughout the entire graph and its derivative is also 1. Power functions [ change | change source ]

Another possibly not so obvious example is the function f ( x ) = 1 x {\displaystyle f(x)={\frac {1}{x}}} . This is essentially the same because 1/x can be simplified to use exponents: f ( x ) = 1 x = x − 1 {\displaystyle f(x)={\frac {1}{x}}=x^{-1}} f ′ ( x ) = − 1 ( x − 2 ) {\displaystyle f'(x)=-1(x^{-2})} f ′ ( x ) = − 1 x 2 {\displaystyle f'(x)=-{\frac {1}{x^{2}}}}

In addition, roots can be changed to use fractional exponents where their derivative can be found: f ( x ) = x 2 3 = x 2 3 {\displaystyle f(x)={\sqrt[{3}]{x^{2}}}=x^{\frac {2}{3}}} f ′ ( x ) = 2 3 ( x − 1 3 ) {\displaystyle f'(x)={\frac {2}{3}}(x^{-{\frac {1}{3}}})} Exponential functions [ change | change source ]

An exponential is of the form a b f ( x ) {\displaystyle ab^{f\left(x\right)}} where a {\displaystyle a} and b {\displaystyle b} are constants and f ( x ) {\displaystyle f(x)} is a function of x {\displaystyle x} . The difference between an exponential and a polynomial is that in a polynomial x {\displaystyle x} is raised to some power whereas in an exponential x {\displaystyle x} is in the power. Example 1 [ change | change source ]

d d x ( 3 ⋅ 2 3 x 2 ) = 3 ⋅ 2 3 x 2 ⋅ 6 x ⋅ ln ⁡ ( 2 ) = ln ⁡ ( 2 ) ⋅ 18 x ⋅ 2 3 x 2 {\displaystyle {\frac {d}{dx}}\left(3\cdot 2^{3x^{2}}\right)=3\cdot 2^{3x^{2}}\cdot 6x\cdot \ln \left(2\right)=\ln \left(2\right)\cdot 18x\cdot 2^{3x^{2}}} Logarithmic functions [ change | change source ]

Take, for example, d d x ln ⁡ ( 5 x ) {\displaystyle {\frac {d}{dx}}\ln \left({\frac {5}{x}}\right)} . This can be reduced to (by the properties of logarithms): d d x ( ln ⁡ ( 5 ) ) − d d x ( ln ⁡ ( x ) ) {\displaystyle {\frac {d}{dx}}(\ln(5))-{\frac {d}{dx}}(\ln(x))}

For derivatives of logarithms not in base e like d d x ( log 10 ⁡ ( x ) ) {\displaystyle {\frac {d}{dx}}(\log _{10}(x))} , this can be reduced to: d d x log 10 ⁡ ( x ) = d d x ln ⁡ x ln ⁡ 10 = 1 ln ⁡ 10 d d x ln ⁡ x = 1 x ln ⁡ ( 10 ) {\displaystyle {\frac {d}{dx}}\log _{10}(x)={\frac {d}{dx}}{\frac {\ln {x}}{\ln {10}}}={\frac {1}{\ln {10}}}{\frac {d}{dx}}\ln {x}={\frac {1}{x\ln(10)}}} Trigonometric functions [ change | change source ]

The cosine function is the derivative of the sine function, while the derivative of cosine is negative sine (provided that x is measured in radians): d d x sin ⁡ ( x ) = cos ⁡ ( x ) {\displaystyle {\frac {d}{dx}}\sin(x)=\cos(x)} d d x cos ⁡ ( x ) = − sin ⁡ ( x ) {\displaystyle {\frac {d}{dx}}\cos(x)=-\sin(x)} d d x sec ⁡ ( x ) = sec ⁡ ( x ) tan ⁡ ( x ) {\displaystyle {\frac {d}{dx}}\sec(x)=\sec(x)\tan(x)} . Properties of derivatives [ change | change source ]

Derivatives can be broken up into smaller parts where they are manageable (as they have only one of the above function characteristics), for example: d d x ( 3 x 6 + x 2 − 6 ) {\displaystyle {\frac {d}{dx}}(3x^{6}+x^{2}-6)} can be broken up as such: d d x ( 3 x 6 ) + d d x ( x 2 ) − d d x ( 6 ) {\displaystyle {\frac {d}{dx}}(3x^{6})+{\frac {d}{dx}}(x^{2})-{\frac {d}{dx}}(6)} = 6 ⋅ 3 x 5 + 2 x − 0 {\displaystyle =6\cdot 3x^{5}+2x-0} = 18 x 5 + 2 x {\displaystyle =18x^{5}+2x\,} Uses of derivatives [ change | change source ]