Derivatives of Inverse Trig Functions — In this section we give the derivatives of all six inverse trig functions. We show the derivation of the formulas for inverse sine, inverse cosine and inverse tangent. Derivatives of Hyperbolic Functions — In this section we define the hyperbolic functions, give the relationships between them and some of the basic facts involving hyperbolic functions.
We also give the derivatives of each of the six hyperbolic functions and show the derivation of the formula for hyperbolic sine. Chain Rule — In this section we discuss one of the more useful and important differentiation formulas, The Chain Rule. With the chain rule in hand we will be able to differentiate a much wider variety of functions. As you will see throughout the rest of your Calculus courses a great many of derivatives you take will involve the chain rule!
Implicit Differentiation — In this section we will discuss implicit differentiation. Not every function can be explicitly written in terms of the independent variable, e. Implicit differentiation will allow us to find the derivative in these cases. Knowing implicit differentiation will allow us to do one of the more important applications of derivatives, Related Rates the next section.
In related rates problems we are give the rate of change of one quantity in a problem and asked to determine the rate of one or more quantities in the problem. This is often one of the more difficult sections for students. We work quite a few problems in this section so hopefully by the end of this section you will get a decent understanding on how these problems work.
Higher Order Derivatives — In this section we define the concept of higher order derivatives and give a quick application of the second order derivative and show how implicit differentiation works for higher order derivatives.
Logarithmic Differentiation — In this section we will discuss logarithmic differentiation. Logarithmic differentiation gives an alternative method for differentiating products and quotients sometimes easier than using product and quotient rule.
Average vs. Newton, Leibniz, and Usain Bolt Opens a modal. Derivative as a concept Opens a modal. Derivative notation review Opens a modal. Derivative as slope of curve Opens a modal. Derivative as slope of curve Get 3 of 4 questions to level up! Secant lines.
Slope of a line secant to a curve Opens a modal. Secant line with arbitrary difference Opens a modal. Secant line with arbitrary point Opens a modal. Secant line with arbitrary difference with simplification Opens a modal. Secant line with arbitrary point with simplification Opens a modal. Secant lines: challenging problem 1 Opens a modal. Secant lines: challenging problem 2 Opens a modal. Derivative definition.
Formal definition of the derivative as a limit Opens a modal. Formal and alternate form of the derivative Opens a modal. Worked example: Derivative as a limit Opens a modal. Worked example: Derivative from limit expression Opens a modal. Finding tangent line equations using the formal definition of a limit Opens a modal. Limit expression for the derivative of function graphical Opens a modal. Practice Derivative as a limit Get 3 of 4 questions to level up!
Estimating derivatives. Estimating derivatives Opens a modal. Practice Estimate derivatives Get 3 of 4 questions to level up! Differentiability and continuity Opens a modal. Differentiability at a point: graphical Opens a modal.
Differentiability at a point: algebraic function is differentiable Opens a modal. Differentiability at a point: algebraic function isn't differentiable Opens a modal. Proof: Differentiability implies continuity Opens a modal. Practice Differentiability at a point: graphical Get 3 of 4 questions to level up!
Differentiability at a point: algebraic Get 3 of 4 questions to level up! Quiz 1. Power rule. Power rule Opens a modal. Power rule with rewriting the expression Opens a modal. Justifying the power rule Opens a modal. Practice Power rule positive integer powers Get 3 of 4 questions to level up! Power rule with rewriting the expression Get 3 of 4 questions to level up! Derivative rules: constant, sum, difference, and constant multiple.
Basic derivative rules Opens a modal. Basic derivative rules: find the error Opens a modal. Basic derivative rules: table Opens a modal. Justifying the basic derivative rules Opens a modal.
Practice Basic derivative rules: find the error Get 3 of 4 questions to level up! Basic derivative rules: table Get 3 of 4 questions to level up!
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