3 The Derivative and Differentiation Rules Free Online FlashCards

How is $f'(a)$ defined by a limit?

The derivative is the limit
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$f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$
when the limit exists. It measures instantaneous change at $a$.

What does $f'(a)$ represent geometrically?

At $(a,f(a))$, $f'(a)$ is the slope of the tangent line. The tangent-line equation is $y-f(a)=f'(a)(x-a)$.

Does differentiability imply continuity?

Differentiability at a point implies continuity there, but continuity does not always imply differentiability. For example, $f(x)=|x|$ is continuous but not differentiable at $x=0$.

What is the power rule?

The power rule is $\frac{d}{dx}[x^n]=nx^{n-1}$, where the expression is defined.

What is the product rule?

The product rule is $\frac{d}{dx}[f(x)g(x)]=f'(x)g(x)+f(x)g'(x)$.

What is the quotient rule?

The quotient rule is $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{g(x)f'(x)-f(x)g'(x)}{[g(x)]^2}$, for $g(x)\ne 0$.

What is the derivative of $e^x$?

The derivative of $e^x$ is $e^x$ itself: $\frac{d}{dx}[e^x]=e^x$.

What is the derivative of $\log_a x$?

For $a>0$ and $a\ne 1$, $\frac{d}{dx}[\log_a x]=\frac{1}{x\ln a}$.

How does the chain rule differentiate $f(g(x))$?

The chain rule gives $\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$: differentiate the outer function, keep the inner expression, then multiply by the inner derivative.

Differentiate $(3x^2-5)^4$.

Using the chain rule, $\frac{d}{dx}[(3x^2-5)^4]=4(3x^2-5)^3(6x)=24x(3x^2-5)^3$.

How is $y^n$ differentiated implicitly?

When differentiating with respect to $x$, treat $y$ as a function of $x$: $\frac{d}{dx}[y^n]=ny^{n-1}\frac{dy}{dx}$.

Find $\frac{dy}{dx}$ for $x^2+y^2=25$.

Differentiating $x^2+y^2=25$ gives $2x+2y\frac{dy}{dx}=0$, so $\frac{dy}{dx}=-\frac{x}{y}$.