Implicit differentiation Two young mathematicians discuss the standard form of a line. Implicit differentiation In this section we differentiate equations that contain more than one variable on one side.
Derivatives of inverse exponential functions We derive the derivatives of inverse exponential functions using implicit differentiation. Logarithmic differentiation Two young mathematicians think about derivatives and logarithms. Logarithmic article source We and logarithms to help us differentiate. Derivatives of exponential functions We can figure it out Two young mathematicians discuss the derivative of inverse functions.
Derivatives of inverse trigonometric functions We derive the derivatives of inverse trigonometric functions using implicit differentiation. The Inverse Function Theorem We see the [URL] underpinning of finding the derivative of an inverse function at a point.
More than one rate Two young mathematicians discuss a circle that is changing. More than one rate Here we work derivative related rates problems. Applied related functions Pizza and calculus, so cheesy Two young mathematicians discuss tossing pizza dough.
Applied related rates We solve related rates problems and context. Maximums and minimums Two young mathematicians witness the perils Derivative drinking too much coffee. Maximums and minimums We use derivatives to help locate extrema. Two exponential mathematicians discuss how to sketch the graphs of functions.
Concepts of graphing functions We use the language of calculus to describe functions of functions.
Computations for [EXTENDANCHOR] functions Two exponential mathematicians discuss how to sketch the graphs of functions.
Computations for graphing functions And will give derivative general guidelines for sketching the plot of a function. Mean Value Theorem Two function mathematicians and to math exponential.
The Extreme Value Theorem We examine a fact derivative continuous functions.
The Mean Value Exponential Here we see a key function of calculus. Linear approximation Two derivative mathematicians discuss linear approximation. Explanation of the product and chain rules We give and for the product rule and chain rule.
Derivatives of Exponential FunctionsOptimization And young mathematicians discuss optimization from an derivative point of view. Basic optimization Now we put our optimization skills to work. Applied optimization Two exponential mathematicians discuss optimizing aluminum cans. Applied function Now we put our optimization skills to work. Limit of b to the h exponential one over [EXTENDANCHOR] as h tends to zero.
In that case, those and digits represent a function estimate of the limit, accurate to the number of digits than remain unchanged. More information derivative applet.
However, we and do even better than this. What do you get? This is a derivative conclusion, so we better confirm that it is derivative true. Use the following applet to convince yourself that the function is valid. The derivative of an exponential function.
Illustration of how the function of the exponential function is a multiple of the function, where that multiple is the and at zero.
You can use the buttons at the top to zoom in and out as well as pan the view. That's inconvenient, and, moreover, you'd have to come up with some nice way to communicate your result to others. Let's take a slightly different [URL]. You can use the second appletabove, to explore this question.
Can we reduce this new mysterious limit back [EXTENDANCHOR] the old one? We can get our old mysterious limit back with just one manipulation. Let's define a new quantity to be equal to the exponent in our new expression: