That is, you take the derivative term by term. Let's do that with the exponential function. We can calculate the derivative of the left side by applying the rule for the derivative of a sum. That is, the derivative of a sum equals the sum of the derivatives of each term. We obtained a surprising result. The expression for the derivative is the same as the one for the original function. That is. Now you can forget for a while the series expression for the exponential. We only needed it here to prove the result above.
We can now apply that to calculate the derivative of other functions involving the exponential. At first sight it may not be obvious, but this is a composite function. This means we need to apply the chain rule. The outer function is the exponential. Its derivative equals itslef. The inner function is ax:. That was simple. It may take a few more examples to get used to the fact that the derivative of an exponential is the same exponential. To calculate its derivative we apply again the chain rule.
As the outer function is the exponential, its derivative equals itself. Here we have a product, so we must use the product rule. 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. Let's see what I mean. First, we apply the product rule. How do we calculate the derivative of this function? We use a trick that is regularly used when dealing with logarithms.
We can write this function as. You can check that this equality is true by using the definition of logarithm. Now we take advantage of the property of logarithms that allows us to take exponents out of the log sign. We'll see more of these curves in Second Order Differential Equations , in the later calculus section. Disclaimer: IntMath. Problem Solver provided by Mathway. Bio: Hi! I am Ligia, I really enjoy mentoring others and have worked as a math tutor at Stanford.
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Derivative of square root of sine x by first principles. Derivative graphs interactive. Differentiating tanh by Haida [Solved! Name optional. Derivatives of Sin, Cos and Tan Functions 2. Derivatives of Csc, Sec and Cot Functions Differentiation interactive applet - trigonometric functions 3.
Derivatives of Inverse Trigonometric Functions 4. Applications: Derivatives of Trigonometric Functions 5. Derivative of the Logarithmic Function 6. Derivative of the Exponential Function 7. Home » Differentiation of Transcendental Functions » 6. Derivative of the Exponential Function. This is a product. Want better grades on your next Trig Exam?
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