Introduction to Differential Calculus

In this tutorial we shall prove the derivative of the hyperbolic cosine function. Let the function be of the form… Click here to read more

In this tutorial we shall prove the derivative of the hyperbolic tangent function. Let the function be of the form… Click here to read more

In this tutorial we shall prove the derivative of the hyperbolic cotangent function. Let the function be of the form… Click here to read more

In this tutorial we shall prove the derivative of the hyperbolic secant function. Let the function be of the form… Click here to read more

In this tutorial we shall prove the derivative of the hyperbolic cosecant function. Let the function be of the form… Click here to read more

In this tutorial we shall discuss the basic formulas of differentiation for hyperbolic functions. 1. $$\frac{d}{{dx}}\sinh x = \cosh x$$… Click here to read more

Example: Differentiate $${x^3}{\tanh ^2}\sqrt x $$ with respect to $$x$$. Consider the function \[y = {x^3}{\tanh ^2}\sqrt x \] Differentiating… Click here to read more

In this tutorial we shall discuss the derivative of the inverse hyperbolic sine function with an example. Let the function… Click here to read more

In this tutorial we shall discuss the derivative of the inverse hyperbolic cosine function with an example. Let the function… Click here to read more

In this tutorial we shall discuss the derivative of the inverse hyperbolic tangent function with an example. Let the function… Click here to read more

In this tutorial we shall discuss the derivative of the inverse hyperbolic tangent function with an example. Let the function… Click here to read more

In this tutorial we shall discuss the derivative of the inverse hyperbolic secant function with an example. Let the function… Click here to read more

In this tutorial we shall discuss the derivative of the inverse hyperbolic cosecant function with an example. Let the function… Click here to read more

In this tutorial we shall discuss basic formulas of differentiation for inverse hyperbolic functions. 1. $$\frac{d}{{dx}}{\sinh ^{ – 1}}x =… Click here to read more

Example: Differentiate $${\cosh ^{ – 1}}\left( {{x^2} + 1} \right)$$ with respect to $$x$$. Consider the function \[y = {\cosh… Click here to read more

Implicit Function If the independent and the dependent variables are mixed up in such a way that the dependent variable… Click here to read more

Example: Find $$\frac{{dy}}{{dx}}$$, if the given implicit function is \[{x^3} + {y^3} = xy\] We have the given implicit function… Click here to read more

In this tutorial we shall find the higher order derivatives. We have already seen how differentiation is applied to a… Click here to read more

Parametric Function A function in which $$x$$ and $$y$$ are expressed as a function of a third variable is called… Click here to read more

Example: Find the second derivative $${y_2}$$ if $$y = \cos \left( {ax + b} \right)$$. We have the given function… Click here to read more

In this tutorial we shall derive the series expansion of $${e^x}$$ by using Maclaurin’s series expansion function. Consider the function… Click here to read more

In this tutorial we shall derive the series expansion of the trigonometric function sine by using Maclaurin’s series expansion function…. Click here to read more

In this tutorial we shall derive the series expansion of the trigonometric function cosine by using Maclaurin’s series expansion function…. Click here to read more

In this tutorial we shall derive the series expansion of the trigonometric function $$\ln \left( {1 – x} \right)$$ by… Click here to read more

In this tutorial we shall derive the series expansion of the trigonometric function $$\ln \left( {1 + x} \right)$$ by… Click here to read more

In this tutorial we shall derive the series expansion of the trigonometric function $${a^x}$$ by using Maclaurin’s series expansion function…. Click here to read more

In this tutorial we shall derive the series expansion of the trigonometric function $${\tan ^{ – 1}}x$$ by using Maclaurin’s… Click here to read more

In this tutorial we shall derive the series expansion of the hyperbolic sine function by using Maclaurin’s series expansion function…. Click here to read more

In this tutorial we shall derive the series expansion of the hyperbolic cosine function by using Maclaurin’s series expansion function…. Click here to read more

In this tutorial we shall derive the series expansion of $$\sqrt {1 + x} $$ by using Maclaurin’s series expansion… Click here to read more

In this tutorial we shall discuss an example of the slope of a tangent to any curve at some given… Click here to read more

In this tutorial we shall look at the differentials of independent and dependent variables. Some applications of differentials will be… Click here to read more

In this tutorial we shall look at the differentials of independent and dependent variables. Some applications of differentials will be… Click here to read more

In this tutorial we shall look at the use of differentials to approximate the value of $$\cos {44^ \circ }$$…. Click here to read more

In this tutorial we shall look at the use differentials to approximate the value of $$\tan {61^ \circ }$$. The… Click here to read more

In this tutorial we shall develop the differentials to approximate the value of $$\sqrt {49.5} $$. The nearest number to… Click here to read more

In this tutorial we shall develop the differentials to approximate the value of $$\csc {61^ \circ }$$. The nearest number… Click here to read more

In this tutorial we shall discuss the application of differentials to approximate any real problem. Let’s look at an example…. Click here to read more

\[\frac{{{\text{dy}}}}{{{\text{dx}}}}\left( {\text{c}} \right) = 0\] \[\frac{{\text{d}}}{{{\text{dx}}}}\left( {{{\text{x}}^{\text{n}}}} \right) = {\text{n}}{{\text{x}}^{{\text{n – 1}}}}\] \[\frac{{\text{d}}}{{{\text{dx}}}}\left[ {{\text{c}}f\left( {\text{x}} \right)} \right] = {\text{c}}f’\left( {\text{x}}… Click here to read more

Example: Find $$\frac{{{\text{dy}}}}{{{\text{dx}}}}$$ if $${\text{y}} = \left( {2{{\text{x}}^3} – 4{{\text{x}}^2}} \right)\left( {3{{\text{x}}^5} + {{\text{x}}^2}} \right)$$ Solution: We have \[{\text{y}} =… Click here to read more

\[\frac{{\text{d}}}{{{\text{dx}}}}\left( {\sin {\text{x}}} \right) = \cos {\text{x}}\] \[\frac{{\text{d}}}{{{\text{dx}}}}\left( {\cos {\text{x}}} \right) = – \sin {\text{x}}\] \[\frac{{\text{d}}}{{{\text{dx}}}}\left( {\tan {\text{x}}} \right) =… Click here to read more

Example: Differentiate $$\frac{{\sqrt {\sin {\text{x}}} }}{{\sin \sqrt {\text{x}} }}$$with respect to ‘x’. Solution: Let $${\text{y}} = \frac{{\sqrt {\sin {\text{x}}} }}{{\sin… Click here to read more