# Taking derivatives

Contents

The big idea of differential calculus is the concept of the derivative, which essentially gives us the direction, or rate of change, of a function at any of its points. Learn all about derivatives and how to find them here.

What's differential calculus all about? An answer to this question lies just right here.

One way of thinking about the derivative (of a function at a point) is as the slope of the tangent line (to the function's graph at that point). Get comfortable with this approach here.

One way of thinking about the derivative is as instantaneous rate of change. This is quite incredible because rate of change is usually found over a period of time, and not at an instant. Get comfortable with this approach here.

Gain some experience working with secant lines. This will help us on our journey to find a formal definition for the derivative.

The derivative is actually a special kind of limit! This is yet another important stop on our continuing journey towards a formal definition of the derivative.

There are two ways to define the derivative of function f at point x=a. The formal definition is the limit of [f(a+h)-f(x)]/h as h approaches 0, and the alternative definition is the limit of [f(x)-f(a)]/(x-a) as x approaches a. Make introduction with these two definitions.

Learn how we can use their formal definition in order to find the derivatives of specific functions. For example, we find the derivative of f(x)=x² at x=3, or for any x-value.

The derivative of a function isn't necessarily defined at every point. Learn about the conditions for the derivative to exist, and specifically about how continuity fits with this story (spoiler: for a function to be differentiable at a point it must be continuous at that point, but the other way isn't necessary).

This may blow your mind, but the derivative of a function is a function in itself! Get comfortable in thinking about the derivative as a function that is separate from, but tightly related to, its original function.

Review your conceptual understanding of derivatives with some challenge problems.

If you ever tried to find derivatives using their formal definition, you probably know how tedious that may be. Fortunately, we have ways for finding derivatives much quicker, using differentiation rules! Make your first steps in this fascinating world by working with the more basic rules. For example, the derivative of [f(x)+g(x)] is f'(x)+g'(x), and the derivative of k⋅f(x) is k⋅f'(x).

The power rule says that the derivative of xⁿ is n⋅xⁿ⁻¹. It allows us to quickly find the derivative of any polynomial, and it doesn't even stop there! Make introduction with this simple but powerful rule.

Put the power rule to use by differentiating various polynomials.

Learn how the power rule can be used to differentiate some basic rational functions.

Learn how the power rule can be used to differentiate some basic radical functions.

The derivative of sin(x) is cos(x) and the derivative of cos(x) is -sin(x). How convenient! Practice differentiating functions that include sine and cosine.

The derivative of eˣ is eˣ. That's pretty amazing. The derivative of ln(x) is 1/x, which is just as surprising.

Review your understanding of basic differentiation rules and your knowledge of the derivatives of common functions with some challenge problems.

The product rule says that the derivative of the product f(x)g(x) is f'(x)g(x)+f(x)g'(x). This helps us find the derivative of a function which is a product of two other, more basic, functions.

The quotient rule says that the derivative of the quotient f(x)/g(x) is [f'(x)g(x)-f(x)g'(x)]/g²(x). This helps us find the derivative of a function which is a quotient of two other, more basic, functions.

The chain rule says that the derivative of the composite function f(g(x)) is f'(g(x))⋅g'(x). This helps us find the derivative of a composite function. It may be slightly hard to grasp, but its importance cannot be overstated!

Go "behind the scenes" with Sal and learn how the chain rule is proved.

Review your understanding of the product, quotient, and chain rules with some challenge problems.

Solve problems involving the derivatives of rational functions. For example, find f'(1) of f(x)=(x+2)/(x-3).

Solve problems involving the derivatives of radical functions. For example, find g'(5) of g(x)=∛(x²+5x-3).

Solve problems involving the derivatives of trigonometric functions. For example, find h'(π) of h(x)=sin(x²-4x+1).

Solve problems involving the derivatives of exponential functions. For example, find f'(1) of f(x)=2^(x²+x+5).

Solve problems involving the derivatives of logarithmic functions. For example, find g'(4) of g(x)=log(x²-1).

Review your function differentiation skills with some challenge problems.

Some two-variable relationships cannot be turned into a function, like the circle equation x²+y²=4. Implicit differentiation allows us to find the derivative of y with respect to x, even in such equations.

Tackle some more advanced problems of implicit differentiation. For example, find dy/dx for e^(xy²)=x-y.

Implicit differentiation allows us to find the derivatives of arcsin(x), arccos(x), and arctan(x).

Implicit differentiation provides us with the relationship between the derivatives of inverse functions: if f and g are inverse functions, then f’(x)=1/(g’(f(x)). Get comfortable working with this relationship.

Equipped with knowledge about the derivatives of all common functions, evaluate some limits that represent various derivatives.

The derivatives of eˣ and ln(x) may be simple, but proving them is a different story. Learn all about it here.

In logarithmic differentiation, we find the derivative of the natural log of a function instead of the derivative of the function itself. It may surprise you, but this can sometimes actually be easier than regular differentiation, if not the only available option. For example, finding the derivative of xˣ.

Parametric and vector-valued functions basically take one input and return two outputs. How do we differentiate such functions? Find out here.

Review your knowledge of the advanced differentiation topics with some challenge problems.

The derivative is a function, and as such it has its own derivative! The same goes for the derivative of the derivative, and so forth. These are all called higher-order derivatives.

Learn how to find the second derivatives (and any higher-order derivative) of parametric and vector-valued functions.