Overview:

# Part 3: Differentiation from first principles

### Finding the gradient at a point on the curve $$y=x^2$$

Given a curve $$y=\text{f}(x)$$, for certain functions $$\text{f}$$, we can find the derivative or gradient function of the curve. The derivative is a function that allows us to find the gradient at any point on the original curve.

Work through this interactive applet showing how to find the gradient function for the curve $$y=x^2$$.

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#### Vocabulary and notation

The derivative or gradient function is a function that allows us to find the gradient at any point on the original curve.

The process of finding the derivative or gradient function is known as differentiation. When looking for the gradient in the $$x$$-$$y$$ plane, we differentiate with respect to $$x$$ to find the derivative with respect to $$x$$.

The derivative of $$y$$ with respect to $$x$$ is written $$\dfrac{\text{d}y}{\text{d}x}$$. We say this as “$$\text{d}y$$ by $$\text{d}x$$”. In the above example, we saw that differentiating $$x^2$$ with respect to $$x$$ gave us $$2x$$.

We also have some alternative notation we can use when working with functions: the derivative of $$\text{f}(x)$$ with respect to $$x$$ is denoted $$\text{f}'(x)$$. We say this as “$$\text{f}$$ prime of $$x$$” or “$$\text{f}$$ dash of $$x$$”.

Questions
Use the method from the applet above to answer the following.
1. If $$y=x^3$$, find $$\dfrac{\text{d}y}{\text{d}x}$$.
2. If $$y=5x^2$$, find $$\dfrac{\text{d}y}{\text{d}x}$$.
3. If $$y=x^3+5x^2$$, find $$\dfrac{\text{d}y}{\text{d}x}$$.
4. If $$\text{f}(x)=x^4$$, find $$\text{f}'(x)$$.
1. If $$y=x^3$$, $$\dfrac{\text{d}y}{\text{d}x}=3x^2$$.
2. If $$y=5x^2$$, $$\dfrac{\text{d}y}{\text{d}x}=10x$$.
3. If $$y=x^3+5x^2$$, $$\dfrac{\text{d}y}{\text{d}x}=3x^2+10x$$.
4. If $$\text{f}(x)=x^4$$, $$\text{f}'(x)=4x^3$$.