Name: Derivatives of Inverse Trig Functions – PowerPoint
SKU: 63b2e8f454be6c454b98f08e
Price: 39.99 USD
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About this resource

What's inside this resource

PowerPoint presentation, 18 slides explaining how to recognize the derivatives of the standard inverse trigonometric functions.

What's included

A title slide stating the learning objective: to recognise the derivatives of the standard inverse trigonometric functions
A recap slide of the constant, constant multiple, sum and difference, product, quotient and chain rules
A recap slide of the derivatives of sine, cosine, tangent, cotangent, secant and cosecant
The inverse function theorem stated for a function that is both invertible and differentiable
Three worked applications of the theorem to ordinary functions, using the quotient rule and the chain rule
A three-slide derivation of the derivative of arcsine, via cos(sin⁻¹x) and the Pythagorean identity
A three-slide derivation of the derivative of arccosine, following the same route with −sin(cos⁻¹x)
A summary slide naming all six inverse functions with their derivatives, from arcsin to arccsc
Three examples: a linear combination of 4cos⁻¹(x) − 10tan⁻¹(x), a product requiring the product rule with sin⁻¹(x), and a motion question finding v(t) from s(t)

Resource types: Teacher Tools, Lesson Plans

In the classroom

Teaching tips & learning objectives

Learning objectives

Recognise the derivatives of the standard inverse trigonometric functions
State the inverse function theorem and use it to differentiate the inverse of a given function
Derive the arcsine and arccosine derivatives from the definition of each inverse and the Pythagorean identity
Differentiate expressions that combine inverse trigonometric functions with constants and with other functions
Find a velocity function by differentiating a displacement function with the chain rule

Teaching tips

The two recap slides on differentiation rules and trig derivatives come before any new content, so they can be shown or skipped depending on how recently the class met them
Three warm-up applications of the inverse function theorem sit between the theorem and the inverse trig work, which makes them the natural pause point in a lesson
Arcsine and arccosine are derived in full while the other four are given as results on the summary slide - decide in advance whether the class will derive one of the remaining four independently
The title slide carries a date field that displays the day the deck is opened, so check what it shows before projecting
Every slide footer carries the contributor's website address, which is worth knowing if slides are shared with students or reused in a handout

Skills covered

Applying the inverse function theorem - three slides work it on ordinary functions, using the quotient rule and the chain rule, before it is used on any inverse trig function.
Deriving the arcsine derivative - the deck moves from the definition of inverse sine to cos(sin⁻¹x), then uses the Pythagorean identity to reach a form students can actually use.
Deriving the arccosine derivative - the same route is repeated with −sin(cos⁻¹x), which lets the sign difference emerge from the working rather than being asserted.
Differentiating combinations - the examples include the linear combination 4cos⁻¹(x) − 10tan⁻¹(x) and a product that requires the product rule alongside sin⁻¹(x).
Connecting differentiation to motion - the closing example finds a velocity function by differentiating a displacement function s(t) with the chain rule.

Good to know

Questions teachers ask about this resource

Which derivations are worked through in full?

Arcsine and arccosine. Each is given three slides that run from the definition of the inverse, through the awkward first expression the theorem produces, to a usable form via the Pythagorean identity. The remaining four appear on a summary slide of all six results, with a note that they follow from the same theorem.

What does a class need to know before this deck lands?

The standard differentiation rules and the trig derivatives - both are recapped on slides 2 and 3 rather than taught - plus the Pythagorean identity, which the arcsine and arccosine derivations use without re-deriving it, and comfort with function inverses.

How do the three closing examples differ from the earlier worked slides?

The earlier ones demonstrate the inverse function theorem on ordinary functions to show the method. The closing three apply the finished results: differentiating a linear combination of arccos and arctan, handling a product involving arcsin, and finding a velocity function from displacement.