This page is a lookup table of derivatives for the functions you meet most often in engineering math. Each row gives a function f(x) on the left and its derivative f'(x) on the right. The rows are grouped by family, from the constant up to the inverse hyperbolic functions. Where a table image carries an error, the correct form is given in the text under it. The meaning of the derivative itself is on the Differentiation page.

Let's keep one thing in mind while you use the table. Every row assumes that the argument is plain x. For an argument such as 3x or x², you combine the row with the chain rule, and the last section shows how.

• Derivatives of Constant
• Derivatives of Polynomials
• Derivatives of Exponential and Logarithmic functions
• Derivatives of Trigonometric Functions
• Derivatives of Hyperbolic Functions
• How do you use the table for a composite function ?

Derivatives of Constant

A constant function has the same value for every x. Its graph is a horizontal line, so its slope is 0 everywhere. This is the simplest row in the table, and it explains why constants disappear when you differentiate a sum.

Figure 1. The derivative of a constant is 0.

You can check it from the definition. With f(x) = c, the difference f(x+h) - f(x) is c - c = 0 for every h. So the difference quotient is 0, and its limit is 0 too. This is why the +5 in x² - 2x + 5 has no effect on the derivative. It also means that two functions that differ only by a constant have the same derivative. That is where the +C in an indefinite integral comes from.

• A constant has zero slope everywhere : its derivative is 0 for every x.
• Constants disappear on differentiation : f(x) and f(x) + c have the same derivative.

Derivatives of Polynomials

A polynomial is a sum of power terms a xⁿ. The power rule handles one term, and a sum is differentiated term by term. So this single row covers every polynomial.

Figure 2. The power rule. The exponent comes down as a factor, and the new exponent is one less.

The rule holds for any real exponent n, not only for positive integers, wherever xⁿ is defined. For example, with n = -1 the derivative of 1/x is -1/x². With n = 1/2 the derivative of √x is 1/(2√x). For a whole polynomial you apply the rule to each term. So the derivative of x² - 2x + 5 is 2x - 2. This slope is 0 at x = 1, and that is where the curve has its minimum.

• The exponent comes down and drops by one : a xⁿ becomes a n x^n-1.
• Negative and fractional exponents work too : 1/x and √x are x^-1 and x^1/2.
• A zero derivative marks a flat point : setting 2x - 2 = 0 finds the minimum of x² - 2x + 5 at x = 1.

Derivatives of Exponential and Logarithmic functions

Exponential and logarithmic functions describe growth, decay and decibel scales, so they appear everywhere in signal and RF work. The key row is e^x, which is its own derivative. The other rows follow from it, because any base can be written as a power of e.

The table below has six rows. The first is a general exponential with base c. Then come the natural exponential, a logarithm with base c, the natural logarithm, the logarithm of |x|, and x^x.

Figure 3. Exponential and logarithmic functions. The first and the fifth row carry errors, and the correct forms are given below.

• The first row should read a ln c * c^ax : the image writes c^ax ln(c*a), but ln(c*a) is ln c + ln a, not a ln c. Write c^ax = e^{ax ln c} and apply the chain rule. The result is a ln c * c^ax, and for c = e it reduces to a e^ax.
• The base c logarithm is a scaled ln x : log_c x = ln x / ln c, so its derivative is 1/(x ln c), as in the image.
• ln x needs x > 0 : the image gives the condition x ≠ 0. The derivative 1/x is right, but ln x itself exists only for x > 0.
• The fifth row should read 1/x : the derivative of ln|x| is 1/x for every x ≠ 0. The image gives |x|/x², which equals 1/|x|. That is right for x > 0 but has the wrong sign for x < 0. At x = -2, for example, the derivative is -1/2, not +1/2.
• x^x uses the same trick : write x^x = e^{x ln x}. The derivative of x ln x is 1 + ln x, so the result is x^x(1 + ln x), as in the image. It needs x > 0.

So the whole group comes from two facts: e^x is its own derivative, and ln x has the derivative 1/x. When a row looks unfamiliar, rewrite it with e and ln, and apply the chain rule.

Derivatives of Trigonometric Functions

Sine and cosine describe every carrier, phasor and rotation in communication engineering. Their derivatives turn into each other, with one sign change. Every other trigonometric row follows from those two by the quotient rule.

The table below comes in two parts. The first part lists the six trigonometric functions. The second part lists their inverse functions, which you meet when you differentiate an angle or a phase.

Figure 4. The six trigonometric functions. The derivative of sine is cosine, and the derivative of cosine is minus sine.

Figure 5. The six inverse trigonometric functions. Each derivative is an algebraic function of x, with no trigonometric function left in it.

• All angles are in radians : in degrees, the derivative of sin x would carry an extra factor π/180.
• tan x follows from the quotient rule : tan x = sin x / cos x, so its derivative is (cos²x + sin²x)/cos²x = 1/cos²x = sec²x.
• The inverse rows come in pairs with opposite sign : arcsin x + arccos x = π/2 is a constant, so their derivatives cancel. The same holds for arctan and arccot, and for arcsec and arccsc.
• Each inverse row has its own domain : arcsin and arccos need |x| < 1, and arcsec and arccsc need |x| > 1. The |x| in the arcsec and arccsc rows comes from the usual choice of principal value.

For a signal, the chain rule adds the angular frequency. The derivative of sin(ωt) with respect to t is ω cos(ωt). So differentiating a sinusoid scales it by ω and shifts it by 90 degrees. This is why the voltage across an inductor, L di/dt, leads its current by 90 degrees.

• Sine and cosine differentiate into each other : sin becomes cos, and cos becomes -sin.
• Differentiation shifts a sinusoid by 90 degrees : and it scales the amplitude by ω.

Derivatives of Hyperbolic Functions

Hyperbolic functions are built from e^x and e^-x. They appear in transmission line equations, in the shape of a hanging cable and in the tanh activation of neural networks. Their derivatives look like the trigonometric ones, but most of the minus signs are gone.

Again the table comes in two parts. The first part lists the six hyperbolic functions, and the second part lists their inverses.

Figure 6. The six hyperbolic functions. The second row names the wrong function, and the correction is given below.

Figure 7. The six inverse hyperbolic functions. The last row carries a sign error, and the correction is given below.

• The cosh row should read sinh(x) : the derivative of cosh(x) is sinh(x) = (e^x - e^-x)/2. The image writes cosh(x) in front of this expression, but the expression itself is sinh(x).
• sinh and cosh differentiate into each other with no sign change : this is the main difference from sin and cos, and it follows from cosh²x - sinh²x = 1.
• tanh has a derivative written in tanh itself : sech²x = 1 - tanh²x. Neural network code uses this form, because tanh(x) is already computed in the forward pass.
• The arccoth row should read 1/(1 - x²) : the image has an extra minus sign. The derivative of arccoth x is the same expression as for arctanh x. The two differ only in domain: |x| > 1 for arccoth and |x| < 1 for arctanh. At x = 3, for example, the derivative is 1/(1 - 9) = -0.125.
• The other inverse rows have domains too : arccosh needs x > 1, and arcsech needs 0 < x < 1.

How do you use the table for a composite function ?

The table only covers a function of plain x. Real expressions have arguments such as ωt, x² or ax. The chain rule connects them to the table, so you rarely need a new row.

The chain rule says that the derivative of f(g(x)) is f'(g(x)) * g'(x). You look up the outer function f in the table and evaluate it at the inner function g(x). Then you multiply by the derivative of the inner function. Let's apply it to four examples, each checked by symbolic computation.

Products and quotients combine rows in a similar way, through the product rule and the quotient rule. The Differentiation Rule page lists those rules with examples.

• Look up the outer function, then multiply by the inner derivative : that is the whole chain rule.
• A linear inner function only adds a factor : sin(ωt) gives ω cos(ωt), and tanh(ax) gives a sech²(ax).