A student computes the derivative of a polynomial motion function. Given coefficient (a) = 4.2000; exponent (n) = 4.0000; x-value (x) = 3.4000, determine the derivative value (fp).

Start with the thinking

• The governing relation printed in this handbook section is Derivatives — power rule.
• Everything except fp is given, so isolate fp symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The power-rule derivative gives the slope of a polynomial function f(x) = a x^n at any point.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$f'(x) = n a x^{n-1}$$

2. Step 2 — Rearrange symbolically for fp:

   $$fp = n a x^{n-1}$$

3. Step 3

   $$List the givens: coefficient (a) = 4.2000, exponent (n) = 4.0000, x-value (x) = 3.4000$$

4. Step 4 — Substitute the given values:

   $$fp = 4.0000 4.2000 3.4000^{4.0000-1}$$

5. Step 5 — Evaluate:

   $$fp = 660.3$$

6. Step 6 — Check: returning fp = 660.3 to

   $$f'(x) = n a x^{n-1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 1,321 — kept a factor of two that cancels in the correct rearrangement.
• 330.2 — dropped that same factor in the other direction.
• 726.3 — rounded an intermediate value before the final step.

Reference: FE Handbook — Derivatives

An engineer takes the derivative of a cost curve to find the marginal rate. Given exponent (n) = 3.0000; x-value (x) = 1.6000; derivative value (fp) = 235.9, determine the coefficient (a).

Start with the thinking

• The governing relation printed in this handbook section is Derivatives — power rule.
• Everything except a is given, so isolate a symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The power-rule derivative gives the slope of a polynomial function f(x) = a x^n at any point.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$f'(x) = n a x^{n-1}$$

2. Step 2 — Rearrange symbolically for a:

   $$a = \dfrac{f'(x)}{n x^{n-1}}$$

3. Step 3

   $$List the givens: exponent (n) = 3.0000, x-value (x) = 1.6000, derivative value (fp) = 235.9$$

4. Step 4 — Substitute the given values:

   $$a = \dfrac{f'(1.6000)}{3.0000 1.6000^{3.0000-1}}$$

5. Step 5 — Evaluate:

   $$a = 30.7161$$

6. Step 6 — Check: returning a = 30.7161 to

   $$f'(x) = n a x^{n-1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 61.4323 — kept a factor of two that cancels in the correct rearrangement.
• 15.3581 — dropped that same factor in the other direction.
• 33.7878 — rounded an intermediate value before the final step.

Reference: FE Handbook — Derivatives

The derivative of a deflection function locates a point of zero slope. Given coefficient (a) = 4.0000; exponent (n) = 3.0000; derivative value (fp) = 460.5, determine the x-value (x).

Start with the thinking

• The governing relation printed in this handbook section is Derivatives — power rule.
• Everything except x is given, so isolate x symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The power-rule derivative gives the slope of a polynomial function f(x) = a x^n at any point.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$f'(x) = n a x^{n-1}$$

2. Step 2 — Rearrange symbolically for x:

   $$x = \left(\dfrac{f'(x)}{n a}\right)^{1/(n-1)}$$

3. Step 3

   $$List the givens: coefficient (a) = 4.0000, exponent (n) = 3.0000, derivative value (fp) = 460.5$$

4. Step 4 — Substitute the given values:

   $$x = \left(\dfrac{f'(x)}{3.0000 4.0000}\right)^{1/(3.0000-1)}$$

5. Step 5 — Evaluate:

   $$x = 6.1948$$

6. Step 6 — Check: returning x = 6.1948 to

   $$f'(x) = n a x^{n-1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 12.3895 — kept a factor of two that cancels in the correct rearrangement.
• 3.0974 — dropped that same factor in the other direction.
• 6.8142 — rounded an intermediate value before the final step.

Reference: FE Handbook — Derivatives

A student computes the derivative of a polynomial motion function. Given coefficient (a) = 4.3000; exponent (n) = 4.0000; x-value (x) = 3.2000, determine the derivative value (fp).

Start with the thinking

• The governing relation printed in this handbook section is Derivatives — power rule.
• Everything except fp is given, so isolate fp symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The power-rule derivative gives the slope of a polynomial function f(x) = a x^n at any point.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$f'(x) = n a x^{n-1}$$

2. Step 2 — Rearrange symbolically for fp:

   $$fp = n a x^{n-1}$$

3. Step 3

   $$List the givens: coefficient (a) = 4.3000, exponent (n) = 4.0000, x-value (x) = 3.2000$$

4. Step 4 — Substitute the given values:

   $$fp = 4.0000 4.3000 3.2000^{4.0000-1}$$

5. Step 5 — Evaluate:

   $$fp = 563.6$$

6. Step 6 — Check: returning fp = 563.6 to

   $$f'(x) = n a x^{n-1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 1,127 — kept a factor of two that cancels in the correct rearrangement.
• 281.8 — dropped that same factor in the other direction.
• 620.0 — rounded an intermediate value before the final step.

Reference: FE Handbook — Derivatives

An engineer takes the derivative of a cost curve to find the marginal rate. Given exponent (n) = 4.0000; x-value (x) = 1.8000; derivative value (fp) = 137.7, determine the coefficient (a).

Start with the thinking

• The governing relation printed in this handbook section is Derivatives — power rule.
• Everything except a is given, so isolate a symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The power-rule derivative gives the slope of a polynomial function f(x) = a x^n at any point.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$f'(x) = n a x^{n-1}$$

2. Step 2 — Rearrange symbolically for a:

   $$a = \dfrac{f'(x)}{n x^{n-1}}$$

3. Step 3

   $$List the givens: exponent (n) = 4.0000, x-value (x) = 1.8000, derivative value (fp) = 137.7$$

4. Step 4 — Substitute the given values:

   $$a = \dfrac{f'(1.8000)}{4.0000 1.8000^{4.0000-1}}$$

5. Step 5 — Evaluate:

   $$a = 5.9028$$

6. Step 6 — Check: returning a = 5.9028 to

   $$f'(x) = n a x^{n-1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 11.8056 — kept a factor of two that cancels in the correct rearrangement.
• 2.9514 — dropped that same factor in the other direction.
• 6.4931 — rounded an intermediate value before the final step.

Reference: FE Handbook — Derivatives

The derivative of a deflection function locates a point of zero slope. Given coefficient (a) = 2.6000; exponent (n) = 4.0000; derivative value (fp) = 399.0, determine the x-value (x).

Start with the thinking

• The governing relation printed in this handbook section is Derivatives — power rule.
• Everything except x is given, so isolate x symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The power-rule derivative gives the slope of a polynomial function f(x) = a x^n at any point.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$f'(x) = n a x^{n-1}$$

2. Step 2 — Rearrange symbolically for x:

   $$x = \left(\dfrac{f'(x)}{n a}\right)^{1/(n-1)}$$

3. Step 3

   $$List the givens: coefficient (a) = 2.6000, exponent (n) = 4.0000, derivative value (fp) = 399.0$$

4. Step 4 — Substitute the given values:

   $$x = \left(\dfrac{f'(x)}{4.0000 2.6000}\right)^{1/(4.0000-1)}$$

5. Step 5 — Evaluate:

   $$x = 3.3727$$

6. Step 6 — Check: returning x = 3.3727 to

   $$f'(x) = n a x^{n-1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 6.7454 — kept a factor of two that cancels in the correct rearrangement.
• 1.6864 — dropped that same factor in the other direction.
• 3.7100 — rounded an intermediate value before the final step.

Reference: FE Handbook — Derivatives

A student computes the derivative of a polynomial motion function. Given coefficient (a) = 1.9000; exponent (n) = 4.0000; x-value (x) = 3.0000, determine the derivative value (fp).

Start with the thinking

• The governing relation printed in this handbook section is Derivatives — power rule.
• Everything except fp is given, so isolate fp symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The power-rule derivative gives the slope of a polynomial function f(x) = a x^n at any point.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$f'(x) = n a x^{n-1}$$

2. Step 2 — Rearrange symbolically for fp:

   $$fp = n a x^{n-1}$$

3. Step 3

   $$List the givens: coefficient (a) = 1.9000, exponent (n) = 4.0000, x-value (x) = 3.0000$$

4. Step 4 — Substitute the given values:

   $$fp = 4.0000 1.9000 3.0000^{4.0000-1}$$

5. Step 5 — Evaluate:

   $$fp = 205.2$$

6. Step 6 — Check: returning fp = 205.2 to

   $$f'(x) = n a x^{n-1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 410.4 — kept a factor of two that cancels in the correct rearrangement.
• 102.6 — dropped that same factor in the other direction.
• 225.7 — rounded an intermediate value before the final step.

Reference: FE Handbook — Derivatives

An engineer takes the derivative of a cost curve to find the marginal rate. Given exponent (n) = 3.0000; x-value (x) = 1.9000; derivative value (fp) = 289.0, determine the coefficient (a).

Start with the thinking

• The governing relation printed in this handbook section is Derivatives — power rule.
• Everything except a is given, so isolate a symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The power-rule derivative gives the slope of a polynomial function f(x) = a x^n at any point.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$f'(x) = n a x^{n-1}$$

2. Step 2 — Rearrange symbolically for a:

   $$a = \dfrac{f'(x)}{n x^{n-1}}$$

3. Step 3

   $$List the givens: exponent (n) = 3.0000, x-value (x) = 1.9000, derivative value (fp) = 289.0$$

4. Step 4 — Substitute the given values:

   $$a = \dfrac{f'(1.9000)}{3.0000 1.9000^{3.0000-1}}$$

5. Step 5 — Evaluate:

   $$a = 26.6851$$

6. Step 6 — Check: returning a = 26.6851 to

   $$f'(x) = n a x^{n-1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 53.3703 — kept a factor of two that cancels in the correct rearrangement.
• 13.3426 — dropped that same factor in the other direction.
• 29.3536 — rounded an intermediate value before the final step.

Reference: FE Handbook — Derivatives

The derivative of a deflection function locates a point of zero slope. Given coefficient (a) = 1.5000; exponent (n) = 4.0000; derivative value (fp) = 332.3, determine the x-value (x).

Start with the thinking

• The governing relation printed in this handbook section is Derivatives — power rule.
• Everything except x is given, so isolate x symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The power-rule derivative gives the slope of a polynomial function f(x) = a x^n at any point.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$f'(x) = n a x^{n-1}$$

2. Step 2 — Rearrange symbolically for x:

   $$x = \left(\dfrac{f'(x)}{n a}\right)^{1/(n-1)}$$

3. Step 3

   $$List the givens: coefficient (a) = 1.5000, exponent (n) = 4.0000, derivative value (fp) = 332.3$$

4. Step 4 — Substitute the given values:

   $$x = \left(\dfrac{f'(x)}{4.0000 1.5000}\right)^{1/(4.0000-1)}$$

5. Step 5 — Evaluate:

   $$x = 3.8118$$

6. Step 6 — Check: returning x = 3.8118 to

   $$f'(x) = n a x^{n-1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 7.6235 — kept a factor of two that cancels in the correct rearrangement.
• 1.9059 — dropped that same factor in the other direction.
• 4.1929 — rounded an intermediate value before the final step.

Reference: FE Handbook — Derivatives

A student computes the derivative of a polynomial motion function. Given coefficient (a) = 2.2000; exponent (n) = 3.0000; x-value (x) = 3.9000, determine the derivative value (fp).

Start with the thinking

• The governing relation printed in this handbook section is Derivatives — power rule.
• Everything except fp is given, so isolate fp symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The power-rule derivative gives the slope of a polynomial function f(x) = a x^n at any point.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$f'(x) = n a x^{n-1}$$

2. Step 2 — Rearrange symbolically for fp:

   $$fp = n a x^{n-1}$$

3. Step 3

   $$List the givens: coefficient (a) = 2.2000, exponent (n) = 3.0000, x-value (x) = 3.9000$$

4. Step 4 — Substitute the given values:

   $$fp = 3.0000 2.2000 3.9000^{3.0000-1}$$

5. Step 5 — Evaluate:

   $$fp = 100.4$$

6. Step 6 — Check: returning fp = 100.4 to

   $$f'(x) = n a x^{n-1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 200.8 — kept a factor of two that cancels in the correct rearrangement.
• 50.1930 — dropped that same factor in the other direction.
• 110.4 — rounded an intermediate value before the final step.

Reference: FE Handbook — Derivatives