What Are the Derivatives of x^n, sin x, cos x, e^x, and ln x?

The derivatives are $\frac{d}{dx}(x^n)=nx^{n-1}$, $\frac{d}{dx}(\sin x)=\cos x$, $\frac{d}{dx}(\cos x)=-\sin x$, $\frac{d}{dx}(e^x)=e^x$, and $\frac{d}{dx}(\ln x)=\frac{1}{x}$. These core rules belong to IB Mathematics AA SL 5.6 and are required at both SL and HL.

These formulas describe instantaneous rates of change: the derivative gives the gradient of the tangent to a function's graph. For the power function, increasing $n$ makes the gradient depend on one lower power of $x$. The sine and cosine rules reflect how their graphs cycle: the slope pattern of sine matches cosine, while the slope pattern of cosine is the negative of sine. The logarithmic result is defined only on its real domain, and all trigonometric differentiation formulas used in IB calculus assume radians, not degrees. This interpretation helps you check signs before calculating.

The power rule works by multiplying by the exponent and reducing that exponent by one. For example,

$$\frac{d}{dx}(x^5)=5x^4, \qquad \frac{d}{dx}(x^{-2})=-2x^{-3}.$$

It also applies to rational powers:

$$\frac{d}{dx}\left(x^{\frac{1}{2}}\right) =\frac{1}{2}x^{-\frac{1}{2}} =\frac{1}{2\sqrt{x}}.$$

The derivative is valid at points where the original function and derivative are defined.

The trigonometric rules assume radian measure. A common misconception is that differentiating $\cos x$ gives $\sin x$. This is incorrect: its graph has negative gradient immediately after $x=0$, so $\frac{d}{dx}(\cos x)=-\sin x$.

The function $e^x$ is special because its rate of change equals its value. By contrast, $\ln x$ differentiates to $\frac{1}{x}$, and its real-valued domain must remain .

For sums and constant multiples, differentiate term by term:

$$\frac{d}{dx}\left(3x^4-2\sin x+5e^x+\ln x\right) =12x^3-2\cos x+5e^x+\frac{1}{x}.$$

When the input is not simply $x$, combine the basic derivative with the chain rule. For example,

$$\frac{d}{dx}(\sin 3x)=3\cos 3x.$$

The inner derivative, $3$, must not be omitted.

Exam technique: These rules may appear on AA SL or HL Paper 1 or Paper 2. Show each differentiated term clearly, use radians for trigonometric calculus, preserve the negative sign for cosine, and apply the chain, product, or quotient rule when the function’s structure requires it.