Differential calculus5/26/2023 ![]() Investigate how to differentiate a parametrized curve in an equation, how the differentiating implicitly defined functions work in calculus and how to obtain the derivatives of logarithmic functions in differential calculus. You will specifically learn about the power rule, the quotient rule, the product rule, the sum and the different rules of differentiation and the various ways to obtain the derivatives of trigonometric functions in differential calculus. This course will then introduce you to the different differentiation methods in differential calculus. Finally, you will discover the importance and the use of the intermediate value theorem in differential calculus. We outline the different ways of obtaining a one-sided derivative, the concept of continuity and differentiability in calculus. We include how to keep a multiplicative constant in differential calculus. Then we discuss the derivative definition and notation, the usefulness of each notation in a differential equation, the different notations for and the processes of finding the derivative of a function. Learn about the characteristics of a normal curve in differential calculus and how to calculate it. Next, examine how to obtain the slope of a curve at a point and the slope of the tangent to a curve. This course introduces you to the concept of differential calculus and the various ways to calculate rates of change in calculus. Differential calculus is a subfield of calculus in mathematics. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. We recommend using aĪuthors: Gilbert Strang, Edwin “Jed” Herman Use the information below to generate a citation. Then you must include on every digital page view the following attribution: If you are redistributing all or part of this book in a digital format, Then you must include on every physical page the following attribution: If you are redistributing all or part of this book in a print format, ![]() Want to cite, share, or modify this book? This book uses theĬreative Commons Attribution-NonCommercial-ShareAlike License d d x ( csch x ) = −csch x coth x d d x ( csch x ) = −csch x coth x Inverse Hyperbolic Functionsģ1. ![]() d d x ( coth x ) = − csch 2 x d d x ( coth x ) = − csch 2 xģ0. d d x ( cosh x ) = sinh x d d x ( cosh x ) = sinh xĢ9. ![]() d d x ( sech x ) = −sech x tanh x d d x ( sech x ) = −sech x tanh xĢ8. d d x ( tanh x ) = sech 2 x d d x ( tanh x ) = sech 2 xĢ7. d d x ( sinh x ) = cosh x d d x ( sinh x ) = cosh xĢ6. d d x ( log b x ) = 1 x ln b d d x ( log b x ) = 1 x ln b Hyperbolic FunctionsĢ5. d d x ( b x ) = b x ln b d d x ( b x ) = b x ln bĢ4. d d x ( ln | x | ) = 1 x d d x ( ln | x | ) = 1 xĢ3. d d x ( e x ) = e x d d x ( e x ) = e xĢ2. d d x ( csc −1 x ) = − 1 | x | x 2 − 1 d d x ( csc −1 x ) = − 1 | x | x 2 − 1 Exponential and Logarithmic FunctionsĢ1. d d x ( csc x ) = −csc x cot x d d x ( csc x ) = −csc x cot x Inverse Trigonometric Functionsġ5. d d x ( cot x ) = − csc 2 x d d x ( cot x ) = − csc 2 xġ4. d d x ( cos x ) = − sin x d d x ( cos x ) = − sin xġ3. d d x ( sec x ) = sec x tan x d d x ( sec x ) = sec x tan xġ2. d d x ( tan x ) = sec 2 x d d x ( tan x ) = sec 2 xġ1. d d x ( sin x ) = cos x d d x ( sin x ) = cos xġ0. d d x ( f ( x ) g ( x ) ) = g ( x ) f ′ ( x ) − f ( x ) g ′ ( x ) ( g ( x ) ) 2 d d x ( f ( x ) g ( x ) ) = g ( x ) f ′ ( x ) − f ( x ) g ′ ( x ) ( g ( x ) ) 2Ĩ. d d x ( f ( x ) − g ( x ) ) = f ′ ( x ) − g ′ ( x ) d d x ( f ( x ) − g ( x ) ) = f ′ ( x ) − g ′ ( x )ħ. d d x ( c f ( x ) ) = c f ′ ( x ) d d x ( c f ( x ) ) = c f ′ ( x )Ħ. d d x ( x n ) = n x n − 1, for real numbers n d d x ( x n ) = n x n − 1, for real numbers nĥ. d d x ( f ( x ) g ( x ) ) = f ′ ( x ) g ( x ) + f ( x ) g ′ ( x ) d d x ( f ( x ) g ( x ) ) = f ′ ( x ) g ( x ) + f ( x ) g ′ ( x )Ĥ. d d x ( f ( x ) + g ( x ) ) = f ′ ( x ) + g ′ ( x ) d d x ( f ( x ) + g ( x ) ) = f ′ ( x ) + g ′ ( x )ģ.
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