About Derivative of e^x
The derivative of e^x with respect to x is e^x. The number e is the base for which the slope of b^x at 0 equals 1. Because of the exponent law, the slope at every point then equals the height. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.
The calculator below opens with this problem already worked, and the full steps are written out above. Change the function to try your own, raise the order for the second or third derivative, or keep the point to see the tangent line.
At x = 0 the slope of e^x is 1, so the function is rising there.
How to use Derivative of e^x
- Type the function in the Function box, for example x^2*sin(x), or tap keys on the key pad. The preview underneath shows how it was read.
- Choose the order: 1st for the ordinary derivative, up to 5th for higher derivatives.
- To differentiate with respect to another letter, change "With respect to". Other letters are then treated as constants, which gives a partial derivative.
- For an equation such as x^2 + y^2 = 25, switch to Implicit and type both sides with an equals sign.
- Enter a point such as 2 or pi/4 to see the value, the slope and the tangent line there, drawn on the graph.
- Read the steps under the result to see which rule was used at each stage, and copy the answer with the copy button.
Features
- First to fifth derivatives of a function of one variable.
- Partial derivatives: choose any variable and the others are held constant.
- Implicit differentiation of an equation in x and y, solved for dy/dx.
- Value, slope and tangent line equation at a point you choose, with the tangent drawn on the graph.
- Step-by-step working that names the power, product, quotient and chain rules and the standard derivative used.
- A live preview and a math key pad, so input is easy on a phone.
- A zoomable, draggable graph of the function and its derivative that you can save as a PNG.
- Clear errors: input the calculator cannot read is flagged with the position of the problem.
The result and the rule behind it
The number e is the base for which the slope of b^x at 0 equals 1. Because of the exponent law, the slope at every point then equals the height.
d/dx e^x = e^xWorked examples
d/dx e^x gives e^x
The number e is the base for which the slope of b^x at 0 equals 1. Because of the exponent law, the slope at every point then equals the height.
Slope of e^x at x = 0 gives 1
Substitute x = 0 into e^x.
Privacy: how it works
- The derivative is worked out by code running in your browser tab. Your functions are not sent to Ulkaa or to any other server, and there is nothing to sign in to.
Tips
- Use * between factors when you can: 2*x is always read as a product.
- To check a result, integrate it with the integral calculator and see if you get the original function back, up to a constant.
- A tangent line is the best straight-line approximation to the function near the point, which is why it is shown next to the graph.
- For the nth derivative of a function that repeats, such as sin(x), pick the order and watch the pattern: sin, cos, -sin, -cos.
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Your functions are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Your functions are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
What is the derivative of e^x?
It is e^x. The number e is the base for which the slope of b^x at 0 equals 1. Because of the exponent law, the slope at every point then equals the height.
What is the slope of e^x at x = 0?
Substituting x = 0 into e^x gives 1, so the function is rising there.
Why is e^x its own derivative?
It is the defining property of e. For any other base b the derivative is b^x times ln(b), and ln(e) = 1 removes the extra factor.
More Derivative Calculator pages
- Derivative CalculatorType a function, get its derivative with every rule named, a tangent line at any point and a graph.
- Derivative of ln(x)Step-by-step derivative of ln(x), with the rule named at each line and a graph of the function and its slope.
- Derivative of sin(x)Step-by-step derivative of sin(x), with the rule named at each line and a graph of the function and its slope.
- Derivative of cos(x)Step-by-step derivative of cos(x), with the rule named at each line and a graph of the function and its slope.
- Derivative of tan(x)Step-by-step derivative of tan(x), with the rule named at each line and a graph of the function and its slope.
- Derivative of sec(x)Step-by-step derivative of sec(x), with the rule named at each line and a graph of the function and its slope.
- Derivative of x^2Step-by-step derivative of x^2, with the rule named at each line and a graph of the function and its slope.
- Derivative of sqrt(x)Step-by-step derivative of sqrt(x), with the rule named at each line and a graph of the function and its slope.
- Derivative of 1/xStep-by-step derivative of 1/x, with the rule named at each line and a graph of the function and its slope.
- Derivative of arctan(x)Step-by-step derivative of arctan(x), with the rule named at each line and a graph of the function and its slope.
- Derivative of x*ln(x)Step-by-step derivative of x*ln(x), with the rule named at each line and a graph of the function and its slope.
- Derivative of e^(2x)Step-by-step derivative of e^(2x), with the rule named at each line and a graph of the function and its slope.
- Derivative of sin^2(x)Step-by-step derivative of sin^2(x), with the rule named at each line and a graph of the function and its slope.