To alleviate this struggle, we want to learn to quickly categorize functions, know which rule to apply, and even rewrite functions in different forms to make differentiation easier.
The derivative of a function describes the function's instantaneous rate of change at a certain point - it gives us the slope of the line tangent to the function's graph at that point.
Let's explore how to differentiate polynomials using the power rule and derivative properties. We work with the function f (x)=x⁵+2x³-x² and apply the power rule to find its derivative, f' (x)=5x⁴+6x²-2x.
In implicit differentiation, we differentiate each side of an equation with two variables (usually x and y ) by treating one of the variables as a function of the other.
The derivative of a function describes the function's instantaneous rate of change at a certain point. Another common interpretation is that the derivative gives us the slope of the line tangent to the function's graph at that point. Learn how we define the derivative using limits.
The chain rule tells us how to find the derivative of a composite function. Brush up on your knowledge of composite functions, and learn how to apply the chain rule correctly.
In this worked example, we dissect the composite function f (x)=ln (√x) into its parts, ln (x) and √x. By applying the chain rule, we successfully differentiate this function, providing a clear step-by-step process for finding the derivative of similar composite functions.
Using the derivative of eˣ and the chain rule, we unravel the mystery behind differentiating exponential functions. We then apply our newfound knowledge to differentiate the expression 8⋅3ˣ.
