Derivatives of trig, exponential and log functions

Japanese school year: Math III

What you learn

This topic covers differentiation formulas for trigonometric, natural exponential, and logarithmic functions. These mathematical tools are crucial for modeling periodic wave oscillations, radioactive decay, and dynamic continuous processes in the physical sciences. Students should have a solid foundation in the fundamental properties and standard limits of these functions.

Go to practice

Key points

This formula shows that differentiating $\sin x$ gives $\cos x$ . The input angle $x$ must be measured in radians rather than degrees.

(\sin x)' = \cos x

This formula shows that differentiating $\cos x$ results in $-\sin x$ . Note that a negative sign appears, unlike when differentiating sine.

(\cos x)' = -\sin x

This formula shows that the exponential function $e^x$ remains identical when differentiated. The base $e$ (about 2.718) is a unique constant where the curve's slope always equals its current value.

(e^x)' = e^x

This formula shows that the derivative of the natural logarithm $\log x$ (base $e$ ) is $\frac{1}{x}$ . It is valid for all positive numbers satisfying $x > 0$ .

(\log x)' = \frac{1}{x}

Choose a set to practice.

Practice 1Coming soon