We can now apply that to calculate the derivative of other functions involving the exponential. Derivative of Inverse Trigonometric Functions: 32) $$\frac{d}{{dx}}Si{n^{ – 1}}x = \frac{1}{{\sqrt {1 – {x^2}} }},{\text{ }} – 1 < x < 1$$, 33) $$\frac{d}{{dx}}Co{s^{ – 1}}x = \frac{{ – 1}}{{\sqrt {1 – {x^2}} }},{\text{ }} – 1 < x < 1$$, 34) $$\frac{d}{{dx}}Ta{n^{ – 1}}x = \frac{1}{{1 + {x^2}}}$$, 35) $$\frac{d}{{dx}}Co{t^{ – 1}}x = \frac{{ – 1}}{{1 + {x^2}}}$$, 36) $$\frac{d}{{dx}}Se{c^{ – 1}}x = \frac{1}{{x\sqrt {{x^2} – 1} }},{\text{ }}\left| x \right| > 1$$, 37) $$\frac{d}{{dx}}Co{\sec ^{ – 1}}x = \frac{{ – 1}}{{x\sqrt {{x^2} – 1} }},{\text{ }}\left| x \right| > 1$$. 4:57 pm. The inner function is ax: That was simple. problem and check your answer with the step-by-step explanations. Copyright © 2005, 2020 - OnlineMathLearning.com. 4) $$\frac{d}{{dx}}{[f(x)]^n} = n{[f(x)]^{n – 1}}\frac{d}{{dx}}f(x)$$ is the Power Rule for Functions. The formula generally given for Power is: W = V x I or W = I 2 x R or W = V 2 / R. Other basic formulae involving Power are: I = W / V or I = (W / R) 2 V = (W x R) 2 or V = W / I The expression for the derivative is the same as the one for the original function. Now, there are some numbers that cancel out, We obtained a surprising result. Just want to thank and congrats you beacuase this project is really noble. b) f(157). IT CHANGED MY PERCEPTION TOWARD CALCULUS, AND BELIEVE ME WHEN I SAY THAT CALCULUS HAS TURNED TO BE MY CHEAPEST UNIT. We welcome your feedback, comments and questions about this site or page. THANKS FOR ALL THE INFORMATION THAT YOU HAVE PROVIDED. Your next step may be to learn about the derivative of ln(x). The new expression for the exponential function was a series, that is, an infinite sum. The following diagram gives the basic derivative rules that you may find useful: Constant Rule, Constant Multiple Rule, Power Rule, Sum Rule, Difference Rule, Product Rule, Quotient Rule, and Chain Rule. Embedded content, if any, are copyrights of their respective owners. Let's see what I mean. Solution: f ’(x) = f ’(x 1) = 1x 0 = 1 Derivative of a Constant Function. Though not part of the original theory, in later years, we have also attributed the Power factor to Ohm as well. Power is defined as time derivative of the work done on a system. Vhia Berania An infinite polynomial is called a power series. Now this is an exponential function with base e, whose derivative we know how to calculate. First, we apply the product rule, Now, to calculate u', we need to apply the chain rule. The formula for Horsepower calculation: First, we will see some equations to help in calculating the horsepower: Power = \( \frac{work}{time} \) For the electric motors, we can calculate the power or horsepower from the torque and speed. THANKS ONCE AGAIN. Using the power rule formula, we find that the derivative of a function that is a constant would be zero. We use a trick that is regularly used when dealing with logarithms. Let's calculate the derivative of the function. To do that, we identify the two factors, This one requires more attention because we need to apply both the product rule and chain rule. i.e. That is, you take the derivative term by term. Example: Differentiate f(x) = x. The derivative of f(x) = x is f ’(x) = 1. which can also be written as. You may ask, the limit definition is much more compact and simple than that ugly infinite sum, why bother? Here we need to apply the chain rule. If a mechanical system has no losses, then the input power must equal the output power. Please submit your feedback or enquiries via our Feedback page. Using the power rule formula, we find that the derivative of the function f(x) = x would be one. The neat thing about a power series is that to calculate its derivative you proceed just like you would with a polynomial. The Sum Rule can be extended to the sum of any number of functions. Substitute x and y with given point’s coordinates i.e here ‘0’ as x and ‘b’ as y, Your email address will not be published. The constant multiple rule says that the derivative of a constant value times a function is the constant times the derivative of the function. Our first contact with number e and the exponential function was on the page about continuous compound interest and number e. In that page, we gave an intuitive definition of number e, and also an intuitive definition of the exponential function. We apply the power rule to calculate the derivative of each term, We cancel out some of the numbers and we arrive to a surprising result, The derivative of the outer function equals the original function, continuous compound interest and number e. This one shows one of the reasons the natural choice for the base of an exponential function is number e. For any other base, you get that ln(a) littering the expression of its derivative. We only needed it here to prove the result above. That's is. As the outer function is the exponential, its derivative equals itself, Here we have a product, so we must use the product rule.