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Essential Concepts

Derivatives of Trigonometric Functions

• We can find the derivatives of $\sin x$ and $\cos x$ by using the definition of derivative and the limit formulas found earlier. The results are
  $\frac{d}{dx} \sin x= \cos x$  and  $\frac{d}{dx} \cos x=−\sin x$.
• With these two formulas, we can determine the derivatives of all six basic trigonometric functions.

The Chain Rule

• The chain rule allows us to differentiate compositions of two or more functions. It states that for $h(x)=f(g(x))$,
  $h^{\prime}(x)=f^{\prime}(g(x))g^{\prime}(x)$
  • In Leibniz’s notation this rule takes the form
    $\frac{dy}{dx}=\frac{dy}{du} \cdot \frac{du}{dx}$
• We can use the chain rule with other rules that we have learned, and we can derive formulas for some of them.
• The chain rule combines with the power rule to form a new rule:
  If $h(x)=(g(x))^n$, then $h^{\prime}(x)=n(g(x))^{n-1}g^{\prime}(x)$
• When applied to the composition of three functions, the chain rule can be expressed as follows: If $h(x)=f(g(k(x)))$, then $h^{\prime}(x)=f^{\prime}(g(k(x)))g^{\prime}(k(x))k^{\prime}(x)$

Derivatives of Inverse Functions

• The inverse function theorem allows us to compute derivatives of inverse functions without using the limit definition of the derivative.
• We can use the inverse function theorem to develop differentiation formulas for the inverse trigonometric functions.

Implicit Differentiation

• We use implicit differentiation to find derivatives of implicitly defined functions (functions defined by equations).
• By using implicit differentiation, we can find the equation of a tangent line to the graph of a curve.

Derivatives of Exponential and Logarithmic Functions

• On the basis of the assumption that the exponential function $y=b^x, \, b>0$ is continuous everywhere and differentiable at $0$, this function is differentiable everywhere and there is a formula for its derivative.
• We can use a formula to find the derivative of $y=\ln x$, and the relationship $\log_b x=\dfrac{\ln x}{\ln b}$ allows us to extend our differentiation formulas to include logarithms with arbitrary bases.
• Logarithmic differentiation allows us to differentiate functions of the form $y=g(x)^{f(x)}$ or very complex functions by taking the natural logarithm of both sides and exploiting the properties of logarithms before differentiating.

Key Equations

• Derivative of sine function
  $\frac{d}{dx}(\sin x)= \cos x$
• Derivative of cosine function
  $\frac{d}{dx}(\cos x)=−\sin x$
• Derivative of tangent function
  $\frac{d}{dx}(\tan x)=\sec^2 x$
• Derivative of cotangent function
  $\frac{d}{dx}(\cot x)=−\csc^2 x$
• Derivative of secant function
  $\frac{d}{dx}(\sec x)= \sec x \tan x$
• Derivative of cosecant function
  $\frac{d}{dx}(\csc x)=−\csc x \cot x$
• The chain rule
  $\frac{d}{dx}(f(g(x)))=f^{\prime}(g(x))g^{\prime}(x)$
• The power rule for functions
  $\frac{d}{dx}((g(x)^n)=n(g(x))^{n-1}g^{\prime}(x)$
• Derivative of inverse sine function
  $\frac{d}{dx}(\sin^{-1} x)=\dfrac{1}{\sqrt{1-x^2}}$
• Derivative of inverse cosine function
  $\frac{d}{dx}(\cos^{-1} x)=\dfrac{-1}{\sqrt{1-x^2}}$
• Derivative of inverse tangent function
  $\frac{d}{dx}(\tan^{-1} x)=\dfrac{1}{1+x^2}$
• Derivative of inverse cotangent function
  $\frac{d}{dx}(\cot^{-1} x)=\dfrac{-1}{1+x^2}$
• Derivative of inverse secant function
  $\frac{d}{dx}(\sec^{-1} x)=\dfrac{1}{|x|\sqrt{x^2-1}}$
• Derivative of inverse cosecant function
  $\frac{d}{dx}(\csc^{-1} x)=\dfrac{-1}{|x|\sqrt{x^2-1}}$
• Inverse function theorem
  $(f^{-1})^{\prime}(x)=\dfrac{1}{f^{\prime}(f^{-1}(x))}$ whenever $f^{\prime}(f^{-1}(x))\ne 0$ and $f(x)$ is differentiable.
• Power rule with rational exponents
  $\frac{d}{dx}(x^{m/n})=\frac{m}{n}x^{(m/n)-1}$.
• Derivative of the natural exponential function
  $\frac{d}{dx}(e^{g(x)})=e^{g(x)} g^{\prime}(x)$
• Derivative of the natural logarithmic function
  $\frac{d}{dx}(\ln (g(x)))=\dfrac{1}{g(x)} g^{\prime}(x)$
• Derivative of the general exponential function
  $\frac{d}{dx}(b^{g(x)})=b^{g(x)} g^{\prime}(x) \ln b$
• Derivative of the general logarithmic function
  $\frac{d}{dx}(\log_b (g(x)))=\dfrac{g^{\prime}(x)}{g(x) \ln b}$

Glossary

chain rule

the chain rule defines the derivative of a composite function as the derivative of the outer function evaluated at the inner function times the derivative of the inner function

implicit differentiation

is a technique for computing $\frac{dy}{dx}$ for a function defined by an equation, accomplished by differentiating both sides of the equation (remembering to treat the variable $y$ as a function) and solving for $\frac{dy}{dx}$

logarithmic differentiation

is a technique that allows us to differentiate a function by first taking the natural logarithm of both sides of an equation, applying properties of logarithms to simplify the equation, and differentiating implicitly

Study Tips

Derivatives of the Sine and Cosine Functions

• Memorize the derivatives of sine and cosine, but understand the proof
• Visualize the relationship between sine/cosine and their derivatives
• Review prerequisite trigonometric identities and limits

Derivatives of Other Trigonometric Functions

• Memorize the derivatives of all six trigonometric functions
• Practice rewriting complex trigonometric expressions before differentiating
• Review common trigonometric identities, especially Pythagorean identities
• Create a quick reference sheet with trig function values for common angles

Higher-Order Derivatives of Trig Functions

• Memorize the 4-step cycle for both sine and cosine
• Practice quickly calculating remainders for large numbers
• Create visual aids (e.g., circular diagrams) to represent the derivative cycle

Deriving the Chain Rule

• Practice identifying the inner and outer functions in composite functions.
• Memorize the chain rule formula and understand its components.
• When applying the chain rule, always work from the outside in.
• Remember that derivatives are evaluated at functions, not at other derivatives.
• Practice with a variety of composite functions, including trigonometric, exponential, and root functions.
• Visualize the chain rule as a process of “unwrapping” nested functions.
• Pay attention to the domain of each function in the composition to ensure the chain rule is applicable.

Combining the Chain Rule With Other Rules

• Practice recognizing when to use the chain rule in combination with other rules.
• When dealing with complex functions, break them down into simpler parts before applying rules.
• Pay attention to the order of operations when combining rules.
• Remember that these combined rules are just applications of the basic chain rule and other differentiation rules.

Applying the Chain Rule Multiple Times

• Always start differentiating from the outermost function and work inward.
• Keep track of where you are in the composition by using parentheses effectively.
• Remember that each application of the chain rule introduces a new factor in the derivative.
• When dealing with physics problems, clearly identify which function represents position, velocity, or acceleration.

The Chain Rule Using Leibniz’s Notation

• Practice identifying the intermediate variable ($u$) in complex functions.
• Remember that the order of multiplication in Leibniz’s notation matters: $\frac{dy}{du} \cdot \frac{du}{dx}$, not the other way around.
• Always express the final answer in terms of the original variable (usually $x$).

Derivatives of Various Inverse Functions

• Remember that the inverse function theorem allows you to find derivatives without explicitly knowing the inverse function.
• When dealing with roots, rewrite them as rational exponents before differentiating.
• Use the extended power rule to differentiate expressions with rational exponents.

Derivatives of Inverse Trigonometric Functions

• Memorize the derivatives of the six inverse trigonometric functions.
• Remember that $\cos(\sin^{-1} x) = \sqrt{1-x^2}$ for $-1 \leq x \leq 1$.
• Pay attention to sign changes in derivatives (e.g., between $\sin^{-1} x$ and $\cos^{-1} x$).
• Be prepared to simplify complex expressions resulting from these derivatives.

What is Implicit Differentiation?

• Practice identifying implicit and explicit functions.
• Remember to use the chain rule when differentiating terms containing $y$.
• Pay attention to the order of operations when differentiating complex expressions.
• Be comfortable with algebraic manipulation to isolate $\frac{dy}{dx}$.
• Recognize that the final expression for $\frac{dy}{dx}$ may contain both $x$ and $y$.

Finding Tangent Lines Implicitly

• Practice implicit differentiation on various types of equations (circles, ellipses, etc.).
• Remember to substitute the given point into $\frac{dy}{dx}$ to find the slope.
• Pay attention to domain restrictions when working with certain curves.
• Review conic sections and their general forms to recognize common implicit equations.

Derivative of the Exponential Function

• Understand why $e$ is special: it’s the only base where $B'(0) = 1$
• Practice applying the chain rule with $e^x$
• Remember that $e^x$ is defined for all real numbers
• Practice finding derivatives of complex expressions involving $e^x$

Derivative of the Logarithmic Function

• Practice using logarithmic properties to simplify expressions before differentiating
• Remember the relationship between exponential and logarithmic functions
• Visualize the graph of $y = \ln x$ and its derivative $y' = \frac{1}{x}$

Logarithmic Differentiation

• Practice identifying functions that benefit from logarithmic differentiation
• Review and memorize logarithm properties
• Pay attention to the chain rule when differentiating logarithmic expressions
• Remember to multiply by $y$ when solving for $\frac{dy}{dx}$